Advertisements
Advertisements
प्रश्न
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
विकल्प
π/6
π/4
π/2
3π/4
Advertisements
उत्तर
3π/4
The given function is
\[f'\left( x \right) = e^x \cos n x + \sin x e^x \]
\[ \Rightarrow f'\left( c \right) = e^c \cos c + \sin c e^c \]
\[\text{Now }, e^x cos x \text { is continuous and derivable in R } . \]
\[\text { Therefore, it is continuous on } \left[ 0, \pi \right] \text { and derivable on} \left( 0, \pi \right) . \]
\[ \therefore f'\left( c \right) = 0 \]
\[ \Rightarrow e^c \left( \cos c + \sin c \right) = 0\]
\[ \Rightarrow \left( \cos c + \sin c \right) = 0 \left( \because e^c \neq 0 \right)\]
\[ \Rightarrow \tan c = - 1\]
\[ \Rightarrow c = \frac{3\pi}{4}, \frac{7\pi}{4}, . . . \]
\[ \therefore c = \frac{3\pi}{4} \in \left( 0, \pi \right)\]
Thus, \[c = \frac{3\pi}{4} \in \left( 0, \pi \right)\]for which Rolle's theorem holds.
Hence, the required value of c is 3π/4.
APPEARS IN
संबंधित प्रश्न
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 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) = x2 − 8x + 12 on [2, 6] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 5x + 4 on [1, 4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
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] ?
At what point on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
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 theorem \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?
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 + x − 1 on [0, 4] ?
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 ?
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).
Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
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`.
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x 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 :-
What does continuity on a closed interval guarantee?
If a differentiable function has an absolute max or min at an interior point \[c\], what must be true?
Why is \[x=0\] a critical point for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
Which list contains all critical points and endpoints for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
