Advertisements
Advertisements
Question
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Solution
We have
f(x)=2x3−9mx2+12m2x+1
⇒f'(x)=6x2−18mx+12m2
Also, f''(x)=12x−18m
Since, f(x) attains its maximum and minimum values at x = p and x = q, respectively, so f '(p) = 0 and
f '(q) = 0
f '(p) = 0
⇒6p2−18mp+12m2=0
⇒p2−3mp+2m2=0
⇒(p−2m)(p−m)=0
⇒p−2m =0 or p−m=0
⇒p=2m or p=m
Similarly,
f '(q) = 0
⇒q=2m or q=m
Now, consider the following cases:
Case I:
If p = 2m and q = 2m, then
p2=q
⇒4m2=2m
⇒2m2−m=0
⇒m(2m−1)=0
∴m=1/2 (m>0)
But, this gives p = 1 as the point of minima, which is not true.
Case II:
If p = 2m and q = m, then
p2=q
⇒4m2=m
⇒4m2−m=0
⇒m(4m−1)=0
∴m=1/4 (m>0)
But, this gives p = 12 as the point of minima, which is not true.
Case III:
If p = m and q = 2m, then
p2=q
⇒m2=2m
⇒m2−2m=0
⇒m(m−2)=0
∴m=2 (m>0)
For this case, p = 2 and q = 4 are the points of maxima and minima, respectively.
Case IV:
If p = m and q = m, then
p2=q
⇒m2=m
⇒m2−m=0
⇒m(m−1)=0
∴m=1 (m>0)
But, this gives q = 1 as the point of maxima, which is not true.
Hence, the value of m is 2.
APPEARS IN
RELATED QUESTIONS
Differentiate sin2 (2x + 1) ?
Differentiate log7 (2x − 3) ?
If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
Find the second order derivatives of the following function log (log x) ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If x = cos θ, y = sin3 θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
If y = etan x, then (cos2 x)y2 =
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
