Advertisements
Advertisements
प्रश्न
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
पर्याय
2
\[- \frac{1}{3}\]
−2
\[\frac{2}{3}\]
Advertisements
उत्तर
2
Given: \[f\left( x \right) = 2 x^3 - 5 x^2 - 4x + 3\] Differentiating the given function with respect to x, we get
\[f'\left( x \right) = 6 x^2 - 10x - 4\]
\[ \Rightarrow f'\left( c \right) = 6 c^2 - 10c - 4\]
\[ \therefore f'\left( c \right) = 0 \]
\[ \Rightarrow 3 c^2 - 5c - 2 = 0\]
\[ \Rightarrow 3 c^2 - 6c + c - 2 = 0\]
\[ \Rightarrow 3c\left( c - 2 \right) + c - 2 = 0\]
\[ \Rightarrow \left( 3c + 1 \right)\left( c - 2 \right) = 0\]
\[ \Rightarrow c = 2, \frac{- 1}{3}\]
\[ \therefore c = 2 \in \left( \frac{1}{3}, 3 \right)\]
Thus,\[c = 2 \in \left( \frac{1}{3}, 3 \right)\] for which Rolle's theorem holds.
Hence, the required value of c 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) = 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) = 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) = x2 − 4x + 3 on [1, 3] ?
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) = ex cos x on [−π/2, π/2] ?
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) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ π ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x 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 \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
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) = x(x −1) on [1, 2] ?
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 = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
State Lagrange's mean value theorem ?
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 ______.
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.
At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
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.
What is the greatest value of a function on the entire given interval called?
For a continuous function on the closed interval \[a,d\], what does \[f(b)\] represent?
For a continuous function on the closed interval \[a,d\], what does \[f(d)\] represent?
Which points are critical points for a function \[f\]?
Which list contains all critical points and endpoints for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
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 \[-1,1\]?
