Question on Integral of the Function

Here is a different question on Integration which will give you the information about negative derivative and evaluate the Function.

Topic : Finding wrong statement from multichoice answers.

Problem : Suppose that f has a negative derivative for all values of x and that f(1) = 0. Which of the following statements must be FALSE about the function h(x) = ₀∫^x f(t) dt ?

(a) The graph of h has a horizontal tangent at x=1.
(b) h has a local maximum at x=1.
(c) The graph of h has an inflection point at x=1.
(d) The graph of dh/dx crosses the x-axis at x=1.

a. (a)
b. (b)
c. (c)
d. (d)

Solution :

Choice c is correct
f'(x)< 0 for all x,
f(1) = 0
h(x)= ₀∫^x f(t) dt

(a) h'(x)= f(x)
h'(1)= f(1)= 0
=> h has a horizontal tangent at x=1
Hence (a) is true.

(b) since h'(1)= 0
And h"(x)=f'(x)< 0 for all x
=>h"(1)< 0
Hence h(x) has a local maximum at x=1
Hence (b) is true

(c)h'(1)= 0
h"(x)=f'(x)< 0 for all x
Hence, the sign of h'(x) does not change as x moves from left to right of 1
=> x=1 is NOT point of inflection
=> (c) is false

(d) h'(1)= 0
h'(x)= f(x)
h'(1)= f(1)= 0

And h"(x)= f'(x)< 0 for all x
=> h'(x) is decreasing function
=> h'(1-)< 0 and h'(1+)> 0
=> h'(x) crosses x - axis at x = 1
Hence (d) is true.