Advertisements
Advertisements
Question
Write the point where f(x) = x log, x attains minimum value.
Advertisements
Solution
\[\text{ Given:} \hspace{0.167em} f\left( x \right) = x \log_e x\]
\[ \Rightarrow f'\left( x \right) = \log_e x + 1\]
\[\text{ For a local maxima or a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \log_e x + 1 = 0\]
\[ \Rightarrow \log_e x = - 1\]
\[ \Rightarrow x = \frac{1}{e}\]
\[ \Rightarrow f\left( \frac{1}{e} \right) = \frac{1}{e} \log_e \left( \frac{1}{e} \right) = - \frac{1}{e}\]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{1}{x}\]
\[\text { At x } = \frac{1}{e}: \]
\[f''\left( \frac{1}{e} \right) = \frac{1}{\frac{1}{e}} = e > 0\]
\[\text { So }, \left( \frac{1}{e}, - \frac{1}{e} \right)\text { is a point of local minimum } . \]
APPEARS IN
RELATED QUESTIONS
f(x)=sin 2x+5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 5)4.
f(x) = \[\frac{1}{x^2 + 2}\] .
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
`f(x)=xsqrt(1-x), x<=1` .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
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]` .
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
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 window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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 maximum value of f(x) = x1/x.
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
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.
Which of the following graph represents the extreme value:-
