Question

Consider the given function and the given interval. f\(x\) = (x - 7)**2 text(, ) [5 text(, ) 11] (a) Find the average value fave of f on the given interval. fave = 4 Correct: Your answer is correct. (b) Find c such that fave = f(c). (Enter solutions from smallest to largest. If there are any unused answer boxes, enter NONE in the last boxes.)

Answer 1

a. $f$ has an average value on [5, 11] of

$f_{\rm ave}=\displaystyle\frac1{11-5}\int_5^{11}(x-7)^2\,\mathrm dx=\frac{(x-7)^3}{18}\bigg|_5^{11}=\frac{4^3-(-2)^3}{18}=4$

b. The mean value theorem guarantees the existence of $c\in(5,11)$ such that $f(c)=f_{\rm ave}$. This happens for

$(c-7)^2=4\implies c-7=\pm2\implies c=9\text{ or }c=5$