Question

Analyzing a graph (AP Calculus)

I'm given this graph of f' on the closed interval [-2, 6] and I'm told that f' has horizontal tangent lines at x = 2 and x = 4.

The function f is twice differentiable with f(3) = 10.

I'm supposed to find the absolute maximum and absolute minimum of f on the interval, and I'm not sure how to begin. Any help is appreciated!

Answer 1

Rule: The absolute extrema of any graph could occur at one of two places

a) At any critical value (assuming this critical value is a local extrema)

b) At either endpoint

So what you do is locate all the critical values and you determine if there is a sign change in f ' (x) to see if those critical values lead to local extrema. Then you test those x values back in f(x).

You also test the x coordinates of the endpoints.

Whichever gives you the largest output is the absolute maximum

Answer 2

so you're given a graph of f ' (x)

what are the roots of f ' (x) ?

Answer 3

The roots of f'(x) would be at (2, 0) and (5, 0)

Answer 4

yes

Answer 5

But I should mention, I'm given no equation for f(x) or f'(x). I'm only given the graph of f'(x) and the information in the first post

Answer 6

draw a number line with the roots on it

Answer 7

Alright

Answer 8

Now we plug in test points. Plug in something to the left of x = 2

say x = 1

so plug in x = 1 into the f ' (x) function and you'll find the result is negative since the curve is below the graph here

Answer 9

then plug in something between x = 2 and x = 5

say x = 3

this result is also negative



so we do NOT have a local extrema at x = 2. We need f ' (x) to have a sign change for that to happen

Answer 10

now plug in something to the right of x = 5

say x = 6

this is positive since f ' (x) is above the x axis here