## Examples 7.1.20(a):

Define a function

*F(x) = t*for^{2}sin(t) dt*x*in the interval*[a, a + 10]*.- Find
*F(a)* - Find
*F'(x)* - Find
*F''(x)* - Find all critical points of
*F(x)*in*[a, a + 10]*

**1.**Since

*F(x) = t*the value

^{2}sin(t) dt*F(a)*is an integral from

*a*to

*a*. But such an integral is 0 regardless of the integrand. Therefore

*F(a) = 0*.

**2.**
The second part is a direct application of the (second) Fundamental
Theorem of Calculus:

t^{2}sin(t) dt = x^{2}sin(x)

**3.**
Since we have computed the first derivative already, it is easy to compute
the second derivative:

F''(x) = F'(x) = x^{2}sin(x) = 2x sin(x) + x^{2}cos(x)

**4.**To find the critical points of

*F*we need to find the points where

*F*is not differentiable or where

*F'(x) = 0*. We know that

*F*is differentiable on any closed interval so that the critical points are those where

Therefore the critical points areF'(x) = x^{2}sin(x) = 0

*x = 0*and

*x = k*,

*k=1, 2, ...*, or better those points

*k = 0, 1, ...*that are inside the interval

*[a, a+10]*.