To determine

To find:

g''4

Answer

-2

Explanation

1) Concept:

Use the formula of slope of line,

dydx=y2-y1x2-x1

2) Given:

gx=∫0xf(t)dt

3) Calculation:

It is from exercise 9,

gx=∫0xf(t)dt,

Differentiate with respect to  x,

ddxgx=ddx∫0xf(t)dt

By Fundamental Theorem of Calculus,

g'x=fx

Again differentiate

g''x=f'x

Substitute  x=4,

g''4=f'4

So, f'4 is the slope of line segment at x=4

Use the formula of slope,

f'4=y2-y1x2-x1

Take any two points on that line segment

3, 25, -2

f'4=-2-25-3

=-42

=-2

So, g''4=-2

Conclusion:

Therefore,

g''4=-2