Let g(x) = ∫ f(t) dt, where ƒ is the function whose graph is shown. Answer the following questions only on the interval [0, 10].

Question

Let where ƒ is the function whose graph is shown. Answer the following questions only on the interval [0, 10].

1. At what values of x does g have a local maximum?

Answer (separate by commas): x =

2. At what values of x does g have a local minimum?

Answer (separate by commas): x =

3. At what value of x does g have an absolute maximum?

Answer: x =

Answer 1

g(x) = ∫_{(from 0 to x)} f(t)dt

By the Fundamental Theorem of Calculus, g'(x) = f(x)

So, g'(x) = 0 when x = 0, 2, 4, 6, 8, 10.

When x = 2, g'(x) changes sign from + to - (local max)

When x = 4, g'(x) changes sign from - to + (local min)

When x = 6, g'(x) changes sign from + to - (local max)

When x = 8, g'(x) changes sign from - to + (local min)

Answer 2

g(x) is the area under the curve of the function "f". Areas that are above the x-axis are positive and areas below the x-axis are negative. Can you see that the area between 0 and 2 is all positive and then between 2 and 4 it is negative? That means the largest value of the area will occur at 2 meaning 2 is a maximum.

Repeat for the other x-axis crossings.