Advertisements
Advertisements
Question
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
Advertisements
Solution
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
RELATED QUESTIONS
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height
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 − 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) = x(x −2)2 on the interval [0, 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(x) = 2 sin x + sin 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
At what point on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?
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 ?
Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?
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 theorem f(x) = sin x − sin 2x − x on [0, π] ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
Let C be a curve defined parametrically as \[x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}\] . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).
For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is
If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,\[\sqrt{3}\]] is
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
A wire of length 50 m is cut into two pieces. One piece of the wire is bent in the shape of a square and the other in the shape of a circle. What should be the length of each piece so that the combined area of the two is minimum?
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
The least value of the function f(x) = 2 cos x + x in the closed interval `[0, π/2]` is:
The function f(x) = [x], where [x] =greater integer of x, is
