Advertisements
Advertisements
प्रश्न
f(x) =\[x\sqrt{1 - x} , x > 0\].
Advertisements
उत्तर
\[\text { Given }: f\left( x \right) = x\sqrt{1 - x}\]
\[ \Rightarrow f'\left( x \right) = \sqrt{1 - x} - \frac{x}{2\sqrt{1 - x}} = \frac{2 - 3x}{2\sqrt{1 - x}}\]
\[\text { For the local maxima or minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \frac{2 - 3x}{2\sqrt{1 - x}} = 0\]
\[ \Rightarrow x = \frac{2}{3}\]
Since, f '(x) changes from positive to negative when x increases through \[\frac{2}{3}\], x = \[\frac{2}{3}\] is a point of maxima.
The local maximum value of f (x) at x = \[\frac{2}{3}\] is given by \[\frac{2}{3}\sqrt{1 - \frac{2}{3}} = \frac{2}{3\sqrt{3}} = \frac{2\sqrt{3}}{9}\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x)=2x3 +5 on R .
f(x) = (x \[-\] 5)4.
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = (x - 1) (x + 2)2.
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
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]` .
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
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.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
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 ?
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
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 the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and 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 _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
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 = ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
