To determine

To find:

The interval where F is concave downward

Answer

(-1, 1)

Explanation

1) Concept:

i) The Fundamental Theorem of Calculus:

If f is continuous on [a, b], then the function g  defined by

gx=∫axf(t)dt       a≤x≤b

is continuous on [a, b] and differentiable on (a,b), and g'x=f(x)

ii)

If f''x<0 for all x in an interval I, the graph of f is concave downward on I

2) Given:

Fx=∫1xftdt

3) Calculations:

By using Fundamental Theorem of Calculus,

F'x=f(x)

Again differentiating,

F''x=f'(x)

From the given graph in interval -1, 1 he graph of  f  is decreasing hence f’(x)<0 that is  F’’(x)<0 in -1, 1 . Hence by definition of concavity F is concave downward within interval (-1, 1)

Conclusion:

Therefore, the function is concave down on interval (-1, 1).