The graph of the derivative f0 of a function f is showm below. (a) Find (or esti

Question

The graph of the derivative f0 of a function f is showm below.

(a) Find (or estimate) all critical points of f and classify each as a local maximum point of f,

a local minimum point of f, or neither.

(b) Determine the intervals over which f is decreasing.

(c) Determine the intervals over which f is concave upward.

(d) Does f attain an absolute maximum or an absolute minimum value over the interval [0;1).

If so, give the x-coordinate and specify which.

The graph of the derivative f0 of a function f is shown below. Find (or estimate) all critical points of f and classify each as a local maximum point of f, a local minimum point of f, or neither. Determine the intervals over which f is decreasing. Determine the intervals over which f is concave upward. Does f attain an absolute maximum or an absolute minimum value over the interval [0;1). If so, give the x-coordinate and specify which.

Explanation / Answer

PART A

Critical points are when f ' (x) = 0

So about when x = -2, 3

PART B

f is decreasing when f ' is negative

During interval -infinity < x < -2

PART C

f is concave upward when derivative of f ' is positive

Derivative of a positive third degree function is a positve parabola so it is concave up during all real numbers or -infinity < x < infinity

PART D

Since f ' is positive during [0, 1), that means f is increasing during that interval, meaning it would attain a minimum value at x = 0. There is no maximum value because f APPROACHES x = 1, so it will never reach x = 1 as one side of the graph is an open interval. If it was a closed interval [0, 1], there would be a max value at x = 1.

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