Advertisements
Advertisements
Question
The minimum value of x loge x is equal to ____________ .
Options
e
`1/e`
`-1/e`
`2/e`
`-e`
Advertisements
Solution
\[\frac{- 1}{e}\]
\[\text { Here }, \]
\[f\left( x \right) = x \log_e x\]
\[ \Rightarrow f'\left( x \right) = \log_e x + 1\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \log_e x + 1 = 0\]
\[ \Rightarrow \log_e x = - 1\]
\[ \Rightarrow x = e^{- 1} \]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{1}{x}\]
\[ \Rightarrow f''\left( e^{- 1} \right) = e > 0\]
\[\text { So,} x = e^{- 1}\text { is a local minima }. \]
\[\text { Hence, the minimum value of } f\left( x \right) = f\left( e^{- 1} \right) . \]
\[ \Rightarrow e^{- 1} \log_e \left( e^{- 1} \right) = - e^{- 1} = \frac{- 1}{e}\]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x) = (x \[-\] 1) (x+2)2.
f(x) =\[x\sqrt{1 - x} , x > 0\].
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
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.
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.
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
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 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 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.
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
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.
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
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.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the point where f(x) = x log, x attains minimum value.
Write the maximum value of f(x) = x1/x.
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] 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 _______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
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 minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Which of the following graph represents the extreme value:-
