Advertisements
Advertisements
प्रश्न
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is ______.
पर्याय
Maximum
Minimum
Zero
Neither maximum nor minimum
Advertisements
उत्तर
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is maximum.
Explanation:
We have f(x) = 2 sin 3x + 3 cos 3x
f'(x) = 2 cos 3x · 3 – 3 sin 3x·3 = 6 cos 3x – 9 sin 3x
f'(x) = – 6 sin 3x · 3 – 9 cos 3x · 3
= – 18 sin 3x – 27 cos 3x
`"f''"((5pi)/6) = - 18 sin 3((5pi)/6) - 27 cos 3((5pi)/6)`
= `- 18 sin ((5pi)/2) - 27 cos((5pi)/2)`
= `-18 sin(2pi + pi/2) - 27 cos(2pi + pi/2)`
= `-18sin pi/2 - 27 cos pi/2`
= – 18 · 1 – 27 · 0
= – 18 < 0 maxima
Maximum value of f(x) at x = `(5pi)/6`
`"f"((5pi)/6) = 2 sin 3((5pi)/6) + 3 cos 3((5pi)/6)`
= `2 sin (5pi)/2 + 3 cos (5pi)/2`
= `2 sin (2x + pi/2) + 3cos(2pi + pi/2)`
= `2 sin pi/2 + 3 cos pi/2`
= 2
APPEARS IN
संबंधित प्रश्न
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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 cylinder is `4/27 pih^3` tan2α.
f(x) = 3 + (x − 2)2/3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
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 ?
\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for 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 −2)2 on the interval [0, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ 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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
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) = x(x −1) 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 theore f(x) = (x − 1)(x − 2)(x − 3) on [0, 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 \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
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) = sin x − sin 2x − x on [0, π] ?
Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
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 from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
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`.
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`
Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.
The maximum value of sinx + cosx is ______.
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
For a continuous function on the closed interval \[a,d\], what does \[f(b)\] represent?
Which statement expresses the Extreme Value Theorem?
Which points must always be checked when finding absolute extrema on a closed interval \[a,b\]?
How are the absolute maximum and absolute minimum determined after evaluating \[f(x)\] at all critical points and endpoints?
What is \[f(1)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What are the absolute extrema of \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
