Advertisements
Advertisements
प्रश्न
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
Advertisements
उत्तर
We have: f (x) = sinx + `sqrt(3)` cosx
= `2(1/2 sin x + sqrt(3)/2 cos x)`
= `2(cos pi/3 sin x + sin pi/3 cos x)`
= `2 sin (x + pi/3)`
f'(x) = `2cos(x + pi/3)`
f"(x) = `-2sin(x + pi/3)`
`"f''"(x)_(x = pi/6) = - 2 sin (pi/6 + pi/3)`
= `- 2 sin pi/2`
= – 2.1
= – 2< 0 ....(Maxima)
= `- 2 xx sqrt(3)/2`
= `- sqrt(3) < 0` .....(Maxima)
Maximum value of the function at x = `pi/6` is
`sin pi/6 + sqrt(3) cos pi/6 = 1/2 + sqrt(3) * sqrt(3)/2` = 2
Hence, the given function has maximum value at x = `pi/6` and the maximum value is 2.
APPEARS IN
संबंधित प्रश्न
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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 ?
f (x) = 2x2 − 5x + 3 on [1, 3] 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) = (x − 1) (x − 2)2 on [1, 2] ?
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) = x(x −2)2 on the interval [0, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 2x on [0, π/2] ?
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) = \[{e^{1 - x}}^2\] on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
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 = x2 on [−2, 2]
?
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) = x3 − 2x2 − x + 3 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 theore f(x) = (x − 1)(x − 2)(x − 3) on [0, 4] ?
Verify the hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = \[\frac{1}{4x - 1},\] 1≤ x ≤ 4 ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
State Rolle's theorem ?
State Lagrange's mean value theorem ?
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
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`.
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.
Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.
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 ______.
The least value of the function f(x) = 2 cos x + x in the closed interval `[0, π/2]` is:
It is given that at x = 1, the function x4 - 62x2 + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
Which statement expresses the Extreme Value Theorem?
How are the absolute maximum and absolute minimum determined after evaluating \[f(x)\] at all critical points and endpoints?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which critical point is obtained from \[2(8x-1)=0\]?
Which list contains all critical points and endpoints for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
