Advertisements
Advertisements
Question
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
Advertisements
Solution
Given function is
\[f\left( x \right) = x \left( x - 4 \right)^2\] , which can be rewritten as \[f\left( x \right) = x^3 - 8 x^2 + 16x\] .
We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function,
\[f\left( x \right)\] is continuous and derivable on \[\left[ 0, 4 \right]\] .
Also,
\[f\left( 0 \right) = f\left( 4 \right) = 0\]
Thus, all the conditions of Rolle's theorem are satisfied.
Now, we have to show that there exists
\[c \in \left[ 0, 4 \right]\] such that \[f'\left( c \right) = 0\] .
We have
\[f\left( x \right) = x^3 - 8 x^2 + 16x\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 16x + 16\]
\[ \therefore f'\left( x \right) = 0 \]
\[ \Rightarrow 3 x^2 - 16x + 16 = 0\]
\[ \Rightarrow 3 x^2 - 12x - 4x + 16 = 0\]
\[ \Rightarrow 3x\left( x - 4 \right) - 4\left( x - 4 \right) = 0\]
\[ \Rightarrow \left( x - 4 \right)\left( 3x - 4 \right)\]
\[ \Rightarrow x = 4, \frac{4}{3}\]
Thus,
\[c = \frac{4}{3} \in \left( 0, 4 \right) \text { such that } f'\left( c \right) = 0\] .
Hence, Rolle's theorem is verified.
APPEARS IN
RELATED QUESTIONS
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
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 ?
\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] 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 − 8x + 12 on [2, 6] ?
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) = ex cos x on [−π/2, π/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 (x) = log (x2 + 2) − log 3 on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
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] ?
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(x) = 2x − x2 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(x) = x3 − 5x2 − 3x on [1, 3] ?
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 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) ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in 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
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 Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.
The maximum value of sinx + cosx is ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
