Provide an appropriate response.
-Suppose that $f$ has a positive derivative for all values of $x$ and that $f ( 2 ) = 0$. Which of the following statements must be true of the function $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ ?
A) The function g has a local maximum at x = 2.
B) The graph of g has an inﬂection point at x = 2.
C) The function g has a local minimum at x = 2.
D) The graph of g crosses the x-axis at x = 2.

Question

Quizplus · Accepted Answer

Since $f(x)$ has a positive derivative for all values of $x$, it is increasing. This means that for values of $t$ less than 2, the value of $f(t)$ is less than zero, while for values of $t$ greater than 2, the value of $f(t)$ is greater than zero. Therefore, the area under the curve of $f(t)$ from 0 to 2 is greater than the area from 2 to $x$. This means that the function $g(x)$, which is the integral of $f(t)$ from 0 to $x$, is increasing from 0 to 2 and decreasing from 2 to infinity. Therefore, $g(x)$ has a local minimum at $x=2$.

Provide an Appropriate Response fff Has a Positive Derivative for All Values Of

Provide an Appropriate Response $f$ Has a Positive Derivative for All Values Of