Showing posts with label Integral of Function. Show all posts
Showing posts with label Integral of Function. Show all posts

Monday, May 11, 2009

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) = 0x 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)= 0x 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.