# Give an example of a function with a critical point that is neither a maximum nor a minimum....

## Question:

Give an example of a function with a critical point that is neither a maximum nor a minimum. Specify the relevant point(s).

## Critical Points

When the derivative of a function equals zero, the function has a critical point. This is a candidate for the function to have a local maximum or minimum, but that behavior is not guaranteed. We need to test this point using the first or second derivative test to classify the point.

The thought behind using critical points to define maxima and minima comes from the fact that when a function has a local maximum or minimum, the first derivative has a value of zero. However, there could be other reasons why the derivative could be zero. Let's analyze a very simple function: {eq}f(x) = x^3 {/eq}.

First, let's find the derivative of this function, which we can do by applying the power rule.

{eq}f'(x) =3 x^2 {/eq}

Next, let's find the critical points of this function by setting this derivative equal to zero and solving.

{eq}3x^2 = 0\\ x = 0 {/eq}

However, we will find that the function does not, in fact, have a local maximum or minimum at this point. We can see this by applying either the first or second derivative test.

{eq}f'(-1) = 3(-1)^2 = 3\\ f'(1) = 3(1)^2 = 3 {/eq}

Since the derivative doesn't change sign on either side of this critical point, the function does not have a local extrema at the critical point we found. If we were to apply the second derivative test, we would find that the second derivative is zero, which is not positive or negative.

{eq}f''(x) = 6x\\ f''(0) = 6(0) = 0 {/eq}

Further testing would show that this is actually an inflection point of the function. The function has a momentary horizontal tangent at the inflection point, but the function continues to increase over its entire domain.