Use the graph of f '(x) below to find the x values of the relative maximum on the graph of f(x):

Question

Use the graph of f '(x) below to find the x values of the relative maximum on the graph of f(x):

FV AP Calculus AB S1 v14 - Activities - Brave (gyazo.com)

It was not letting me add a picture of the graph.

A. 0.5

B. 2

C. 1

D. 1.5

Answer 1

Since this is a graph of f '(x), we want x-values where f '(x) = 0. From the graph, looks like this happens at x=0, x=1 and x=2, etc. But these are x values for minimums and maximums of f(x) .

f '(x) appears to be 2sin(πx). So f(x) would be -2cos(πx)/π.

f(0) = -2/π

f(.5) = 0

f(1) = 2/π

f(1.5) = 0

f(2) = -2/π

Thus f(1) is a maximum, so answer (C)

Answer 2

To find a relative maximum, you have two conditions:

f'(x) = 0: This gives the x-values that are extrema - either minima, or maxima.

f''(x) < 0: If the first derivative is zero, you apply the second derivative test to distinguish between minima and maxima. If the second derivative is negative, you have a maximum.

So, to apply this to your problem: f'(x) is 0 at x = 0, 1, 2. This narrows down to choices b & c. f''(x) is the derivative of f'(x), and since we only care about the sign of f''(x), you can graphically see whether the slope of f'(x) is positive or negative at these two points.

I'll leave this last step to you :)