Find the critical points of 𝑓 and determine whether they are local minima, local maxima, or neither.

Question

The figure shows the graph of the derivative  of a function  Find the critical points of  and determine whether they are local minima, local maxima, or neither.

https://ibb.co/q0ZFXJm

Answer 1

The critical point is where the first derivative equals 0.

They are -1, .5, and 2

At -1, note how the first derivative is positive on both sides. This means that -1 will be an inflection point rather than a minimum or maximum. At .5, the first derivative is positive to the left and negative to the right, so this is a maximum. At 2, the first derivative is negative to the left and positive to the right, so this is a minimum.

Another way to look at this: the derivative of the derivative, or second derivative, is 0 at -1, negative at .5, and positive at 2, so the function is concave at .5 and convex at 2 so we have a maximum at .5 and minimum at 2. At -1, where the derivative of the derivative is 0, we see that the third derivative is positive (the first derivative is convex), so this is an inflection and not a minimum or maximum.

-1 neither

.5 maximum

2 minimum