Advertisements
Advertisements
प्रश्न
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 ?
Advertisements
उत्तर
As, the Rolle's theorem holds for the function f(x) = x3 + bx2 + cx, x \[\in\] [1, 2] at the point x = \[\frac{4}{3}\]
\[\text { So,} f\left( 1 \right) = f\left( 2 \right)\]
\[ \Rightarrow \left( 1 \right)^3 + b \left( 1 \right)^2 + c\left( 1 \right) = \left( 2 \right)^3 + b \left( 2 \right)^2 + c\left( 2 \right)\]
\[ \Rightarrow 1 + b + c = 8 + 4b + 2c\]
\[ \Rightarrow 3b + c + 7 = 0 . . . . . \left( i \right)\]
\[\text { And } f'\left( \frac{4}{3} \right) = 0\]
\[ \Rightarrow 3 \left( \frac{4}{3} \right)^2 + 2b\left( \frac{4}{3} \right) + c = 0 \left[ As, f'\left( x \right) = 3 x^2 + 2bx + c \right]\]
\[ \Rightarrow \frac{16}{3} + \frac{8b}{3} + c = 0\]
\[ \Rightarrow 8b + 3c + 16 = 0 . . . . . \left( ii \right)\]
\[\left( ii \right) - \left( i \right) \times 3, \text { we ge }\]
\[8b - 9b + 16 - 21 = 0\]
\[ \Rightarrow - b - 5 = 0\]
\[ \Rightarrow b = - 5\]
\[\text { Substituting b } = - 5 \text { in} \left( i \right), \text { we get }\]
\[3\left( - 5 \right) + c + 7 = 0\]
\[ \Rightarrow - 15 + c + 7 = 0\]
\[ \Rightarrow c = 8\]
APPEARS IN
संबंधित प्रश्न
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 ______.
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 − 4)2 on the interval [0, 4] ?
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 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) = sin 2x on [0, π/2] ?
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) = 2 sin x + sin 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
At what point on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 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(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) = 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 theorem f(x) = x2 − 2x + 4 on [1, 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) = 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) = x2 + x − 1 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] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
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 ______.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is ______.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
The minimum value of `1/x log x` in the interval `[2, oo]` is
For a continuous function on the closed interval \[a,d\], what does \[f(a)\] represent?
For a continuous function on the closed interval \[a,d\], what does \[f(d)\] represent?
After finding critical points and identifying endpoints, what should be done next?
What is \[f(-1)\] 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 \[-1,1\]?
