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Both the concavity and convexity can occur in a function once or more than once. The point where the function is neither concave nor convex is known as inflection point or the point of inflection. In this article, the concept and meaning of inflection point, how to determine the inflection point graphically are explained in detail.

The point of inflection or inflection point is a point in which the concavity of the function changes. It means that the function changes from concave down to concave up or vice versa. In other words, the point in which the rate of change of slope from increasing to decreasing manner or vice versa is known as an inflection point. Those points are certainly not local maxima or minima. They are stationary points.

Generally, when the curve of a function bends, it forms a concave shape. It is known as the concavity of a function. In graph function, two types of concavity can be found.

Concave Up – If a curve opens in an upward direction or it bends up to make a shape like a cup, it is said to be concave up or convex down.

Concave Down – If a curve bends down or resembles a cap, it is known as concave down or convex up. In other words, the tangent lies underneath the curve if the slope of the tangent increases by the increase in an independent variable.

If f(x) is a differentiable function, then f(x) is said to be:

Here, f “(x) is the second order derivative of the function f(x).

The point of inflection defines the slope of a graph of a function in which the particular point is zero. The following graph shows the function has an inflection point.

It is noted that in a single curve or within the given interval of a function, there can be more than one point of inflection.

An inflection point is defined as a point on the curve in which the concavity changes. (i.e) sign of the curvature changes. We know that if f ” > 0, then the function is concave up and if f ” < 0, then the function is concave down. If the function changes from positive to negative, or from negative to positive, at a specific point x = c, then that point is known as the point of inflection on a graph.

We can identify the inflection point of a function based on the sign of the second derivative of the given function.  Also, by considering the value of the first-order derivative of the function, the point inflection can be categorized into two types, as given below.

If f'(x) is equal to zero, then the point is a stationary point of inflection.

If f'(x) is not equal to zero, then the point is a non-stationary point of inflection.

Click here to get the inflection point calculator.

Refer to the following problem to understand the concept of an inflection point.

Example:

Determine the inflection point for the given function f(x) = x4 – 24x2+11

Solution:

Given function: f(x) = x4 – 24x2+11

The first derivative of the function is

f’(x) = 4x3 – 48x

The second derivative of the function is

f”(x) = 12x2 – 48

Set f”(x) = 0,

12x2 – 48 = 0

Divide by 12 on both sides, we get

x2 – 4 = 0

x2 = 4

Therefore, x = ± 2

To check or x = 2, substitute x= 1 and 3 in f”(x)

So, f”(1) = 12(1)2 – 48 = -36 (negative)

f”(3) = 12(3)2 – 48 = 276 (positive)

To check for x = -2, substitute x= 0 and -3 in f”(x)

So, f”(0) = 12(0)2 – 48 = -48 (negative)

f”(3) = 12(3)2 – 48 = 276 (positive)

Hence, proved

Now, substitute x = ± 2 in f”(x)

Therefore, it becomes

f”(2) = 12(2)2 – 48 = -69

f”(-2) = 12(-2)2 – 48 = -69

Therefore, the inflection points are (2, -69), and (-2, -69).

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Deep brain stimulation (DBS) is the use of a surgically implanted device that delivers mild electrical pulses to the fornix, an area of the brain that plays a central role in memory. Currently, DBS is FDA approved for the treatment of disabling symptoms of Parkinson’s disease and other neurologic conditions. In this study, we are looking to evaluate the effectiveness of deep brain stimulation of the fornix to slow the cognitive decline associated with Alzheimer’s. Individual with a diagnosis of dementia or mild Alzheimer’s disease, is 65 years or older, and is willing to undergo neurosurgery may be eligible for the study.

The study will involve 12 clinic visits over 48 months at Johns Hopkins Bayview Campus. Participants in the trial will be compensated for their time and transportation. To hear more information regarding this study, please call us at (410) 550-4969 or email jkim587@jhmi.edu. Principle investigator: Paul Rosenberg, M.D. (IRB00181686).

Apathy is one of the most prevalent neuropsychiatric symptoms in individuals with Alzheimer’s disease.Up to 50% of people with dementia experience this loss of will and initiative, lack of interest in activities,lack of productivity, or lack of response to events in their life. The goal of the ADMET II study is to see ifmethylphenidate can help lessen the effects of apathy in Alzheimer’s disease. Also known as Ritalin, methylphenidate is approved for use in the treatment of attention Deficit/Hyperactivity Disorder by the FDA, and previous trials have suggested it may be helpful in treating apathy in Alzheimer’s disease. Individuals in the ADMET II trial will be asked to take methylphenidate or placebo for 6 months and attend about 7 clinic visits at Johns Hopkins at Bayview. If you are interested in learning more about this trial, call 410-550-9022. Principal Investigator: Paul Rosenberg, M.D. (JHU protocol number IRB00064958)

The combination of memory problems and abnormal blood sugar has been shown to be a risk factor for developing Alzheimer’s disease, and a nutraceutical (“health food”) is being investigated to target this problem. This trial is seeking individuals between the ages of 50-90 who are concerned that they may be experiencing memory loss. BDPP is a combination of resveratrol (a chemical found in the skin of grapes) and grape seed extract. Study investigators hope to learn if BDPP could help maintain memory or decrease the amount of abnormal proteins in the brain. Individuals in this study will be asked to take BDPP or placebo for 4 months and attend about 9 research visits to Johns Hopkins at Bayview. Glucose testing and a memory evaluation are provided as part of the trial. Please call 410-550-6486 for more information on how to enroll. Principal investigator: Paul Rosenberg, M.D. (IRB protocol number IRB00062802)

Agitation in Alzheimer’s disease can be challenging for affected individuals and the people who care for them. Some researchers think that agitation in Alzheimer’s disease may be caused by changes in a person’s normal sleep cycle. Individuals with these changes may become more likely to become restless, wander and pace, resist help with daily activities, or become hard to handle at certain times in the day. The goal of the Actigraphy in Agitation study is to learn more about changes in sleep patterns that occur in people with Alzheimer’s disease. The Actigraphy in Agitation study is looking for people who have Alzheimer’s disease with or without symptoms of agitation including pacing, resisting care, and restlessness. Participants in the Actigraphy in Agitation study will be asked to attend 1 clinic visit at Johns Hopkins at Bayview, wear a device the size of a wrist watch on their wrist for 7 days and use two devices to help researchers learn about their sleep patterns. The study team will provide all of the required equipment at no cost. If you are interested in learning more about this trial, please contact John Outen at 410-550-7385 or jouten3@jhmi.edu. Principal Investigator: Paul Rosenberg, M.D. (JHU protocol number IRB00151016). Download flyer.

Agitation is common in people who have Alzheimer’s disease (AD) and can be can be a challenging behavior for the family and friends of these individuals. Agitation affects 42-60 percent of people with Alzheimer’s Disease, and involves emotional distress, excess psychomotor activity, aggression, or disinhibition. Treatment options for agitation in AD are limited. Current approaches are inadequate and better approaches are needed. S-CitAD is a placebo-controlled, masked, randomized clinical trial, sponsored by the National Institute on Aging (NIA). The study is designed to examine the efficacy and safety of escitalopram in combination with a psychosocial intervention (PSI) as treatment for agitation in AD patients. Individuals in the S-CitAD trial will be enrolled for 24-weeks, with treatment for 12 weeks, and additional 12 weeks of safety and efficacy follow-up, for a total of 6 clinic visits at Johns Hopkins University. If you are interested in learning more about this trial, call (410) 550-9022. Site Investigator: Paul Rosenberg, MD. (Principal Investigator: Constantine Lyketsos, MD; Protocol number: IRB00148995)

Are you over 60 and feeling depressed? Symptoms of depression in older adults are common yet often go undetected. Symptoms could include feelings of sadness or hopelessness, loss of energy, inability to enjoy pleasurable activities, changes in appetite or sleeping patterns, or poor concentration/memory. If you are feeling depressed, not taking antidepressant medication and in good physical health you may be eligible to participate in a research study involving treatment. Qualified people will participate at no cost to them and will be compensated for their time and transportation. For more information about the research study, please call us at 410-550-4192 or email us at BrainImagingStudy@lists.johnshopkins.edu . Principal Investigator: Gwenn Smith, PhD (IRB Protocol No: NA_00021615) Download PDF flyer

The FOXY trial is for people between the ages of 30-80 who have frontotemporal dementia (FTD). The FOXY trial aims to help researchers learn if giving Oxytocin to people who have FTD may be helpful in reducing social apathy, indifference, and lack of empathy. These traits are common in people who have FTD and can be burdensome for family and friends. You may qualify for this study if you have been diagnosed FTD, display symptoms of moderate social apathy/indifference, and have someone willing to act as a study partner.

People who join this trial will be given Oxytocin or placebo for about 18 weeks. There will be 5 clinic visits and 2 telephone calls that will occur will taking study drug (or placebo). During 4 of the visits, we will ask you to eat a meal at the clinic while we film you. Please call (410)-550-6486 or e-mail twhite46@jhmi.edu for more information on how to enroll. Principal Investigator: Chiadi Onyike, M.D. (IRB protocol number IRB00146529)

The Johns Hopkins Memory Center is currently conducting a study of the differences between normal aging, mild memory problems, and the onset of memory disorders like Alzheimer’s Disease. If you choose to participate in the Memory and Aging Study, you and a study partner will be invited to our clinic for a 2-3 hour assessment including memory testing, physical exam, and blood samples. We will ask you and your study partner questions about your daily functioning. You will return to our clinic once a year for an annual physical exam and memory testing. People, 60 years old or older, with or without memory problems, can participate in this study. If you are interested in learning more about this study please call Irene Giorgakis at 410-550-9021. Constantine Lyketsos, MD, MPH, Principal Investigator (JHU IRB Application No: NA_00045104)

Are you more forgetful lately? Do you have trouble with the names of people you’ve met recently? Do you get lost in new places? Do you have a greater tendency to misplace things? Do other people notice that you are forgetful? If you are age 55 or over, having memory problems, not taking antidepressant medication and in good health, you may be eligible to participate in a research study. Qualified people will participate at no cost to them and will be compensated for their time and transportation. For more information about the research study, please call us at 410-550-4192 or email us at BrainImagingStudy@lists.johnshopkins.edu. (Principal Investigator: Gwenn Smith, PhD, IRB Protocol No: NA_00026190) Download flyer.

Managing money and personal finances are daily activities for most people. Individuals with Mild Cognitive Impairment (MCI) may have difficulty managing their money and making financial decisions. This may lead to poor outcomes, including increase in conflict within families, financial victimization, and the onset of dementia. Researchers in the Neural Network Connectivity of Financial Capacity Study want to learn more about how brain changes affect financial skills through the use of Magnetic Resonance Imaging (MRI). Individuals with or without MCI greater than 65 years old may be eligible. Participants in the NCFC study will be asked to attend 1 clinic visit at Johns Hopkins at Bayview for neurocognitive testing and 1 visit at the Kennedy Krieger Institute in East Baltimore for a state-of-the-art MRI of the brain. If you are interested in learning more about this trial, please contact John Outen at 410-550-7385 or jouten3@jhmi.edu. Principal Investigator: Milap Nowrangi, M.D. (JHU protocol number IRB00129816). Download flyer.

The purpose of this research is to find if there is a relationship between a chemical in the blood, which may be related to memory problems, and the body’s hormonal response from the pancreas, gut, and fat tissue after an oral glucose tolerance test. People with and without memory problems may join this study. The study involves a memory screening test and an oral glucose tolerance test. The results of these tests will help us with our research study related to memory, aging, and hormones. If you are interested in learning more about this study please call 410-550-4969. Esther Oh, M.D., Principal Investigator (JHU IRB Application No.: NA_00014837)

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