Advertisements
Advertisements
Question
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
Advertisements
Solution
f(x)=sin2x-cosx
f'(x)=2 sinx.cosx+sinx
=sinx(2cosx+1)
Equating f’(x) to zero.
f'(x)=0
sin x(2cos x + 1) = 0
sin x = 0
∴ x = 0, π
`2cos x + 1 = 0`
`⇒cos x =-1/2`
`therefore x=(5pi)/6`
`f(0) = sin20 – cos 0 = − 1`
`f((5pi)/6)=sin^2(5pi/6)-cos((5pi)/6)`
`=sin^2(pi/6)+cos(pi/6)`
`=1/4-sqrt3/2`
`=((1-2sqrt3)/sqrt4)`
`f(pi) = sin^2pi – cospi = 1`
Of these values, the maximum value is 1, and the minimum value is −1.
Thus, the absolute maximum and absolute minimum values of f(x) are 1 and −1, which it attains at x = 0 and x = π.
APPEARS IN
RELATED QUESTIONS
f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 8x + 12 on [2, 6] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = (x2 − 1) (x − 2) on [−1, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [−π/4, π/4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex cos x on [−π/2, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 2x + 4 on [1, 5] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore f(x) = tan−1 x on [0, 1] ?
State Rolle's theorem ?
If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?
For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is
The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
The minimum value of `1/x log x` in the interval `[2, oo]` is
The function f(x) = [x], where [x] =greater integer of x, is
For a continuous function on the closed interval \[a,d\], what does \[f(c)\] represent?
After finding critical points and identifying endpoints, what should be done next?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which derivative is correct?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f(0)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
