Advertisements
Advertisements
Question
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Solution
\[\text { Given: } f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right)^2 \]
\[ = \left( x - 1 \right)\left( x^2 - 4x + 4 \right)\]
\[ = x^3 - 4 x^2 + 4x - x^2 + 4x - 4\]
\[ = x^3 - 5 x^2 + 8x - 4\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 10x + 8\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 10x + 8 = 0\]
\[ \Rightarrow 3 x^2 - 6x - 4x + 8 = 0\]
\[ \Rightarrow \left( x - 2 \right)\left( 3x - 4 \right) = 0\]
\[ \Rightarrow x = 2 \text { and }\frac{4}{3}\]
\[\text { Thus, x = 2 and } x = \frac{4}{3} \text { are the possible points of local maxima or local minima } . \]
\[\text { Now }, \]
\[f''\left( x \right) = 6x - 10\]
\[At x = 2: \]
\[ f''\left( 2 \right) = 6\left( 2 \right) - 10 = 2 > 0\]
\[\text { So, x = 2 is the point of local minimum }. \]
\[\text { The local minimum value is given by }\]
\[f\left( 2 \right) = \left( 2 - 1 \right) \left( 2 - 2 \right)^2 = 0\]
\[\text { At }x = \frac{4}{3}: \]
\[ f''\left( \frac{4}{3} \right) = 6\left( \frac{4}{3} \right) - 10 = - 2 < 0\]
\[\text { So, x} = \frac{4}{3}\text { is the point of local maximum } . \]
\[\text { The local maximum value is given by }\]
\[f\left( \frac{4}{3} \right) = \left( \frac{4}{3} - 1 \right) \left( \frac{4}{3} - 2 \right)^2 = \frac{1}{3} \times \frac{4}{9} = \frac{4}{27}\]
APPEARS IN
RELATED QUESTIONS
f(x) = 4x2 + 4 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 5)4.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = (x - 1) (x + 2)2.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum and minimum values of y = tan \[x - 2x\] .
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
Write sufficient conditions for a point x = c to be a point of local maximum.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
For the function f(x) = \[x + \frac{1}{x}\]
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
The minimum value of x loge x is equal to ____________ .
Which of the following graph represents the extreme value:-
