CPM Homework Help : CALC3RD Problem 8-105

Multiple Choice: If $F ( x ) = \int _ { 0 } ^ { x } \sqrt { 9 - t } d t$ , which of the following statements are true?

$F^\prime(5) = 2$

$F(−7) > F(5)$

$F$ is concave downward

I only

II only

I and III

II and III

I and II

Hint 1:

Recall that the derivative of an integral gives you the original function (FTC, part 1).

Hint 2:

$\text{Visualize the graph of }y=\sqrt{9-x}.$

$\text{Without computing, compare }F(-7)=\int_{0}^{-7}\sqrt{9-x}dx\text{ with }F(5)=\int_{0}^{5}\sqrt{9-x} dx.$

Which is greater?

Hint 3:

The derivative of the integral, $F\left(x\right)$ , gives you the original function, $f\left(x\right)$ .
So, the derivative of the original function will give you the second derivative of $F\left(x\right)$ .