Advertisements
Advertisements
Question
Write the minimum value of f(x) = xx .
Advertisements
Solution
\[\text { Given: } \hspace{0.167em} f\left( x \right) = x^x \]
\[\text { Taking log on both sides, we get }\]
\[\log f\left( x \right) = x \log x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{1}{f\left( x \right)} f'\left( x \right) = \log x + 1\]
\[ \Rightarrow f'\left( x \right) = f\left( x \right) \left( \log x + 1 \right)\]
\[ \Rightarrow f'\left( x \right) = x^x \left( \log x + 1 \right) .............. \left( 1 \right)\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow x^x \left( \log x + 1 \right) = 0\]
\[ \Rightarrow \log x = - 1\]
\[ \Rightarrow x = \frac{1}{e}\]
\[\text { Now }, \]
\[f''\left( x \right) = x^x \left( \log x + 1 \right)^2 + x^x \times \frac{1}{x} = x^x \left( \log x + 1 \right)^2 + x^{x - 1} \]
\[\text { At }x = \frac{1}{e}: \]
\[f''\left( \frac{1}{e} \right) = \frac{1}{e}^\frac{1}{e} \left( \log\frac{1}{e} + 1 \right)^2 + \frac{1}{e}^\frac{1}{e} - 1 = \frac{1}{e}^\frac{1}{e} - 1 > 0\]
\[\text { So,} x = \frac{1}{e}\text { is a point of local minimum .} \]
\[\text { Thus, the minimum value is given by }\]
\[f\left( \frac{1}{e} \right) = \frac{1}{e}^\frac{1}{e} = e^\frac{- 1}{e} \]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x)=2x3 +5 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
`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) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
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.
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?
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
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 \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
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.
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 ?
A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
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 necessary condition for a point x = c to be an extreme point of the function f(x).
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
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 maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
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?
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.
