Let `f(x)=int_(0)^(x) (sint-cost)(e^t-2)(t-1)^3(t-1)^3(t-2)^5 dt , 0lt xle4`
Then , the number of points where f(x) assumes local maximum value , is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

C

Explanation

We have `f(x)=oversetxunderset0int(sint-cost)(e^t-2)(t-1)^3(t-1)^3(t-2)^5 dt , 0lt xle4`
`rArr f'(x)=oversetxunderset0int(sint-cost)(e^t-2)(x-1)^3(x-2)^5,0ltxle4`
for local maximum or minimum , we must have
f'(x)=0
`rArr sinx-cosx=0,e^x-2=0,(x-1)^3=0,(x-2)^5=0`
`rArr tanx=1,e^x=2,x=1,2`
`rArr x=5/4,(5pi)/4,x=log_e2,x=1,2 " " [therefore0 lt xle 4]`
`rArrx=0.785,3.925,0.693,1,2`
The changes in sign s of f'(x) in the nieghbourhoods of these points are shown in Fig.17.
Clearly `x=log_e2,pi/4 and (5pi)/4` are points of local maxima