Thread: Find Max Value Based on F' Graph

The graph of f', the derivative of the function f, is shown above for x on the closed interval of [0,10]. The areas of the regions between the graph of f' and the x-axis are 20, 6, and 4, respectively. If f(0)=2, what is the maximum value of f on the closed interval [0,10]?

a. 16
b. 20
c. 22
d. 30
e. 32

I have no clue how to solve this one.

Originally Posted by summermagic

The graph of f', the derivative of the function f, is shown above for x on the closed interval of [0,10]. The areas of the regions between the graph of f' and the x-axis are 20, 6, and 4, respectively. If f(0)=2, what is the maximum value of f on the closed interval [0,10]?

a. 16
b. 20
c. 22
d. 30
e. 32

f has maximums where f' changes sign from positive to negative and at the right endpoint, x = 10.

let the first x-intercept > 0 be $\displaystyle a$

$\displaystyle f(a) = 2 + \int_0^a f'(x) \, dx = 2 + 20 = 22$

$\displaystyle f(10) = 2 + \int_0^{10} f'(x) \, dx = 2 + 18 = 20$

maximum value of f(x) on the interval [0, 10] should now be clear.