Advertisements
Advertisements
प्रश्न
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
उत्तर
\[\text { Given }: \hspace{0.167em} f\left( x \right) = x + \frac{1}{x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]
\[\text { For a local maxima or a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]
\[ \Rightarrow x^2 = 1\]
\[ \Rightarrow x = 1, - 1\]
\[\text { But }x > 0\]
\[ \Rightarrow x = 1\]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{1}{x^3}\]
\[\text { At x} = 1: \]
\[f''\left( 1 \right) = \frac{2}{\left( 1 \right)^3} = 2 > 0\]
\[\text { So, x = 1 is a point of local minimum } . \]
\[\text { Thus, the local minimum value is given by }\]
\[f\left( 1 \right) = 1 + \frac{1}{1} = 1 + 1 = 2\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = - (x-1)2+2 on R ?
f(x)=| x+2 | on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
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) = 2/x - 2/x^2, x>0`
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
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 \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
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 ?
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
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 minimum value of f(x) = xx .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Which of the following graph represents the extreme value:-
