Advertisements
Advertisements
प्रश्न
Write the minimum value of f(x) = xx .
उत्तर
\[\text { Given: } \hspace{0.167em} f\left( x \right) = x^x \]
\[\text { Taking log on both sides, we get }\]
\[\log f\left( x \right) = x \log x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{1}{f\left( x \right)} f'\left( x \right) = \log x + 1\]
\[ \Rightarrow f'\left( x \right) = f\left( x \right) \left( \log x + 1 \right)\]
\[ \Rightarrow f'\left( x \right) = x^x \left( \log x + 1 \right) .............. \left( 1 \right)\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow x^x \left( \log x + 1 \right) = 0\]
\[ \Rightarrow \log x = - 1\]
\[ \Rightarrow x = \frac{1}{e}\]
\[\text { Now }, \]
\[f''\left( x \right) = x^x \left( \log x + 1 \right)^2 + x^x \times \frac{1}{x} = x^x \left( \log x + 1 \right)^2 + x^{x - 1} \]
\[\text { At }x = \frac{1}{e}: \]
\[f''\left( \frac{1}{e} \right) = \frac{1}{e}^\frac{1}{e} \left( \log\frac{1}{e} + 1 \right)^2 + \frac{1}{e}^\frac{1}{e} - 1 = \frac{1}{e}^\frac{1}{e} - 1 > 0\]
\[\text { So,} x = \frac{1}{e}\text { is a point of local minimum .} \]
\[\text { Thus, the minimum value is given by }\]
\[f\left( \frac{1}{e} \right) = \frac{1}{e}^\frac{1}{e} = e^\frac{- 1}{e} \]
APPEARS IN
संबंधित प्रश्न
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 \[-\] 3x .
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
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]` .
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
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.
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
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 point on the curve x2 = 8y which is nearest to the point (2, 4) ?
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) = x1/x.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
The minimum value of x loge x is equal to ____________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
