Advertisements
Advertisements
प्रश्न
`f(x) = 2/x - 2/x^2, x>0`
उत्तर
\[\text { Given }: f\left( x \right) = \frac{2}{x} - \frac{2}{x^2} = 2 x^{- 1} - 2 x^{- 2} \]
\[ \Rightarrow f'\left( x \right) = - 2 x^{- 2} + 4 x^{- 3} = \frac{4}{x^3} - \frac{2}{x^2}\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow \frac{4}{x^3} - \frac{2}{x^2} = 0\]
\[ \Rightarrow 4 - 2x = 0\]
\[ \Rightarrow x = 2\]
\[\text { Thus, x = 2 is the possible point of local maxima or local minima }. \]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{- 12}{x^4} + \frac{4}{x^3}\]
\[\text { At }x = 2: \]
\[ f''\left( 2 \right) = \frac{- 12}{16} + \frac{4}{8} = \frac{- 12 + 8}{16} = \frac{- 1}{4} < 0\]
\[\text { So, x = 2 is the point of local maximum }. \]
\[\text { The local maximum value is given by }\]
\[f\left( 2 \right) = \frac{2}{2} - \frac{2}{2^2} = 1 - \frac{1}{2} = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) = xex.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
`f(x)=xsqrt(1-x), x<=1` .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
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 maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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 sufficient conditions for a point x = c to be a point of local maximum.
The maximum value of x1/x, x > 0 is __________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of the sum of their volumes.
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
