Advertisements
Advertisements
Question
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
Advertisements
Solution
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
RELATED QUESTIONS
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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 ______.
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height
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 ?
\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for each of the following function on the indicated interval f (x) = cos 2 (x − π/4) 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) = cos 2x on [0, π] ?
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(x) = sin x − sin 2x on [0, π]?
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 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 theorem f(x) = x2 − 3x + 2 on [−1, 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 \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
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) = x2 + x − 1 on [0, 4] ?
Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
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) ?
Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
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 Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x 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
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = `pi/6`
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.
The maximum value of sinx + cosx is ______.
The least value of the function f(x) = 2 cos x + x in the closed interval `[0, π/2]` is:
The minimum value of `1/x log x` in the interval `[2, oo]` is
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
