Home
Class 12
MATHS
If sinx+xge|k|x^(2), AA x in [0,(pi)/(2)...

If `sinx+xge|k|x^(2), AA x in [0,(pi)/(2)]`, then the greatest value of k is

A

`(-2(2+pi))/(pi^(2))`

B

`(2(2+pi))/(pi^(2))`

C

can't be determined finitely

D

zero

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)=sinx+x,`
`rArr" "f'(x)=cos x+1 gt 0`
`rArr" "f(x)` is increasing function
Also `f''(x)=-sinx lt0`
`therefore" f is concave downward for " x in [0,(pi)/(2)]`
Now `g(x)=|k|x^(2)` is concave upward and increasing
So, if `g((pi)/(2))le f((pi)/(2))`, then `f(x)geg(x)`
`rArr" "1+(pi)/(2)ge|k|(pi^(2))/(4)`
`rArr" "k=[(-(2pi+4))/(pi^(2)),((2pi+4))/(pi^(2))]`
Promotional Banner

Topper's Solved these Questions

  • MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS

    CENGAGE|Exercise Multiple Correct Answer Type|10 Videos

  • MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS

    CENGAGE|Exercise Comprehension Type|6 Videos

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE|Exercise JEE Advanced Previous Year|17 Videos

  • PAIR OF STRAIGHT LINES

    CENGAGE|Exercise Exercise (Numerical)|5 Videos

Similar Questions

Explore conceptually related problems

Prove that sinx+2xgeq(3x(x+1))/pi, AA x in [0,pi/2] (Justify the inequality, if any used).

The range of f(x)=[1+sinx]+[2+sin(x/2)]+[3+sin (x/3)]+....+[n+sin (x/n)]AAx in [0,pi] , where [.] denotes the greatest integer function, is,

Let g:[-2,2] rarr R , where g(x)=x^(2015)+sgn(x)+[(x^2+1)/p] be an odd function for all x in[-2,2] then the smallest integral value of p is equal to '[Note: [k] denote the greatest integer less than or equal to k .]

If |cosec x|=(5pi)/(4)+(x)/(2)AA x in(-2pi,2pi) , then the number of solutions are

If x^(2)-6x+k = 0 has equal roots the value of k is

If [(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]^k=[(1,0),(0,1)], then the least positive integral value of k , is

Two parallel chords of a circle of radius 2 are at a distance. sqrt(3+1) apart. If the chord subtend angles pi/k and (2pi)/k at the center, where k >0, then the value of [k] is

If the focus of the parabola x^2-k y+3=0 is (0,2), then a values of k is (are) 4 (b) 6 (c) 3 (d) 2

If the equation sin ^(2) x - k sin x - 3 = 0 has exactly two distinct real roots in [0, pi] , then find the values of k .

Let f(x) = |(2cos^2x, sin2x, -sinx), (sin2x, 2 sin^2x, cosx), (sinx, -cosx,0)| , then the value of int_0^(pi//2){f(x) + f'(x)} dx is