Advertisements
Advertisements
प्रश्न
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
Advertisements
उत्तर
\[\text { Given }: \hspace{0.167em} f\left( x \right) = \frac{x}{2} + \frac{2}{x}\]
\[ \Rightarrow f'\left( x \right) = \frac{1}{2} - \frac{2}{x^2}\]
\[\text { For the local maxima or minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \frac{1}{2} - \frac{2}{x^2} = 0\]
\[ \Rightarrow \frac{1}{2} = \frac{2}{x^2}\]
\[ \Rightarrow x^2 = \pm 2\]
Since x > 0, f '(x) changes from negative to positive when x increases through 2. So, x = 2 is a point of local minima.
The local minimum value of f (x) at x = 2 is given by \[\frac{2}{2} + \frac{2}{2} = 2\]
APPEARS IN
संबंधित प्रश्न
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
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) = 2x3 − 15x2 + 36x + 1 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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
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 \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
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 total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
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.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
