Advertisements
Advertisements
प्रश्न
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
Advertisements
उत्तर
We have
`f(x)=sinx−cosx 0<x<2π `
`f'(x)=ddx(sinx−cosx) `
`=cosx+sinx`
For maxima and minima, we have
`f'(x)=0`
`⇒cosx+sinx=0`
`⇒cosx=−sinx`
`⇒x=(3π)/4,(7π)/4`
Now,
`f"(x)=d/dx(cosx+sinx) `
`=−sinx+cosx`
`"At " x=(3π)/4`
`f"((3π)/4)=−sin((3π)/4)+cos((3π)/4)`
`=-1/sqrt2-1/sqrt2`
`=-sqrt2`
`⇒f"((3π)/4)<0`
Thus, `x=(3π)/4` is the point of local maxima.
Local maximum value `f((3π)/4)`
`=sin((3π)/4)−cos((3π)/4)`
`=1/sqrt2+1/sqrt2=sqrt2`
`At x=(7π)/4`
`f"((7π)/4)=−sin((7π)/4)+cos((7π)/4)`
`=1/sqrt2+1/sqrt2=sqrt2`
`⇒f"((7π)/4)>0`
Thus, `x=(7π)/4` is the point of local minima.
Local minimum value of `f(x)=f((7π)/4)`
`sin((7π)/4)-cos((7π)/4)`
`=-1/sqrt2-1/sqrt2`
`=-sqrt2`
APPEARS IN
संबंधित प्रश्न
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of ______.
f (x) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 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 − 4x + 3 on [1, 3] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 1)2 on [0, 1] ?
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) = 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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
It is given that the Rolle's theorem holds for the function f(x) = x3 + bx2 + cx, x \[\in\] at the point x = \[\frac{4}{3}\] , Find the values of b and c ?
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 − 1 on [2, 3] ?
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 \[f\left( x \right) = \sqrt{25 - x^2}\] on [−3, 4] ?
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] ?
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 \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Verify the hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = \[\frac{1}{4x - 1},\] 1≤ x ≤ 4 ?
Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
