Home
Class 12
MATHS
Consider the function f(x)=int(0)^(x)(5l...

Consider the function `f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt`. `f(x)` is

A

negative for all `x in (0,1)`

B

increasing for all `x in (0,1)`

C

decreasing for all `x in (0,1)`

D

non-monotonic function for `x in (0,1)`

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt.`
`rArr" "f'(x)=5ln(1+x^(2))-10x tan^(-1)x+16 sinx`
`rArr" "f''(x)=2(8 cos x-5 tan^(-1)x)`
`rArr" "f''(x)=-2(8sinx+(5)/(1+x^(2)))lt0AAx in (0,1)`
So, f''(x) is decreasing `AA x in (0,1)`
`rArr" "f''(x)gtf''(1)=2(8cos1-(5pi)/(4))`
`" "gt2(8cos.(pi)/(3)-(5pi)/(4))`
`" "=2(4-(5pi)/(4))gt0`
So, f''(x) is increasing, for `x gt 0 , f'(x)gtf'(0)=0`
So, f(x) is increasing, for `x gt0, f(x) gt f(0)=0`
So, `int_(0)^(x)f(t)` is positive and increasing.
Promotional Banner

Similar Questions

Explore conceptually related problems

Consider the function f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt . Which is not true for int_(0)^(x)f(t)dt gt?

If f(x)=int_(2)^(x)(dt)/(1+t^(4)) , then

In (-4,4) the function f(x)=int_(-10)^x (t^2-4)e^(-4t) dt , has

For the functions f(x)= int_(0)^(x) (sin t)/t dt where x gt 0 . At x=n pi f(x) attains

If f(x)=x^(2)int_(0)^(1)f(t)dt+2 , then

(i) If f(x) = int_(0)^(sin^(2)x)sin^(-1)sqrt(t)dt+int_(0)^(cos^(2)x)cos^(-1)sqrt(t) dt, then prove that f'(x) = 0 AA x in R . (ii) Find the value of x for which function f(x) = int_(-1)^(x) t(e^(t)-1)(t-1)(t-2)^(3)(t-3)^(5)dt has a local minimum.

Find the points of minima for f(x)=int_0^x t(t-1)(t-2)dt

Find the points of minima for f(x)=int_0^x t(t-1)(t-2)dt

" If " f(x) =int_(0)^(x)" t sin t dt tehn " f(x) is

If f(x)=int_(1)^(x)(logt)(1+t+t^(2))dt AAxge1 , then prove that f(x)=f(1/x) .