Answer
Interval $(-1,2)$ is the interval $(a,b)$ that allows for a maximum value in our original integral.
Work Step by Step
Find the interval $(a,b)$ for which $\int^b_a2+x-x^2dx$ is a maximum
Step 1
Define f as $f(x)=\int_a^x2+t-t^2dt$ our new goal is to find the maximum value of this function. We will do this by finding the critical points.
Step 2
Take the derivate of $f(x)$ by the Fundamental Theorem of Calculus #1.
$f'(x)=2+x-x^2$
Step 3
Find the critical points of $f'(x)$ to find the maximum values.
$2+x-x^2 = 0$
$x^2-x-2 = 0$
$(x+1)(x-2)=1$
The critical points of $f(x)$ are -1, and 2.
Step 4
Test whether $f(x)$ is decreasing or increasing via sample points on the intervals that separate the critical points $(-\infty,-1), (-1,2), (1,\infty)$
$(-\infty,-1)$ is decreasing
$(-1,2)$ is increasing
$(2,\infty)$ is decreasing
This means that the interval $(-1,2)$ is the interval $(a,b)$ that allows for a maximum value in our original integral.
