Advertisements
Advertisements
प्रश्न
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
पर्याय
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)
f (x) = 2x2 − 5x + 3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
f (x) = x2/3 on [−1, 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) = x(x − 1)2 on [0, 1] ?
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) = 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 \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
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] ?
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 theorem f(x) = x(x + 4)2 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 theorem f(x) = sin x − sin 2x − x on [0, π] ?
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 ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?
State Rolle's theorem ?
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
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
The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,\[\sqrt{3}\]] 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.
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
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 ______.
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
