Advertisements
Advertisements
प्रश्न
f(x) = (x \[-\] 1) (x \[-\] 2)2.
उत्तर
\[\text { Given: } f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right)^2 \]
\[ = \left( x - 1 \right)\left( x^2 - 4x + 4 \right)\]
\[ = x^3 - 4 x^2 + 4x - x^2 + 4x - 4\]
\[ = x^3 - 5 x^2 + 8x - 4\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 10x + 8\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 10x + 8 = 0\]
\[ \Rightarrow 3 x^2 - 6x - 4x + 8 = 0\]
\[ \Rightarrow \left( x - 2 \right)\left( 3x - 4 \right) = 0\]
\[ \Rightarrow x = 2 \text { and }\frac{4}{3}\]
\[\text { Thus, x = 2 and } x = \frac{4}{3} \text { are the possible points of local maxima or local minima } . \]
\[\text { Now }, \]
\[f''\left( x \right) = 6x - 10\]
\[At x = 2: \]
\[ f''\left( 2 \right) = 6\left( 2 \right) - 10 = 2 > 0\]
\[\text { So, x = 2 is the point of local minimum }. \]
\[\text { The local minimum value is given by }\]
\[f\left( 2 \right) = \left( 2 - 1 \right) \left( 2 - 2 \right)^2 = 0\]
\[\text { At }x = \frac{4}{3}: \]
\[ f''\left( \frac{4}{3} \right) = 6\left( \frac{4}{3} \right) - 10 = - 2 < 0\]
\[\text { So, x} = \frac{4}{3}\text { is the point of local maximum } . \]
\[\text { The local maximum value is given by }\]
\[f\left( \frac{4}{3} \right) = \left( \frac{4}{3} - 1 \right) \left( \frac{4}{3} - 2 \right)^2 = \frac{1}{3} \times \frac{4}{9} = \frac{4}{27}\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = - (x-1)2+2 on R ?
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 5)4.
f(x) = sin 2x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x3\[-\] 6x2 + 9x + 15
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
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 necessary condition for a point x = c to be an extreme point of the function f(x).
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
The maximum value of x1/x, x > 0 is __________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Which of the following graph represents the extreme value:-
