Answer
See explanation.
Work Step by Step
By theorem 5.5.6, we then know that:
\[
f(x) \leq M \Rightarrow -M+f(x) \leq 0
\]
$\int_{a}^{b}[-M+f(x)] d x \leq 0$
$\Rightarrow -\int_{a}^{b} M d x+\int_{a}^{b} f(x) d x \leq 0$
$\Rightarrow \int_{a}^{b} f(x) d x-M(-a+b) \leq 0$
$\Rightarrow \int_{a}^{b} f(x) d x \leq (-a+b)M$
By theorem 5.5.6, we then know that:
$f(x) \geq m \Rightarrow -m+f(x) \geq 0$
$\int_{a}^{b}[f(x)-m] d x \geq 0$
$\Rightarrow -\int_{a}^{b} m d x +\int_{a}^{b} f(x) d x\geq 0$
$\Rightarrow -m(-a+b) +\int_{a}^{b} f(x) d x\geq 0$
$\Rightarrow \int_{a}^{b} f(x) d x \geq (-a+b)m$
Combining these two parts, we obtain:
$(-a+b)m \leq \int_{a}^{b} f(x) d x \leq (-a+b)M$