Advertisements
Advertisements
प्रश्न
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
Advertisements
उत्तर
\[\text { Given }: \hspace{0.167em} f\left( x \right) = \frac{\log x}{x}\]
\[ \Rightarrow f'\left( x \right) = \frac{1 - \log x}{x^2}\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \frac{1 - \log x}{x^2} = 0\]
\[ \Rightarrow 1 - \log x = 0\]
\[ \Rightarrow \log x = 1\]
\[ \Rightarrow \log x = \log e\]
\[ \Rightarrow x = e\]
\[\text { Now,} \]
\[f''\left( x \right) = \frac{- x - 2x\left( 1 - \log x \right)}{x^4} = \frac{- 3x - 2x \log x}{x^4}\]
\[\text { At }x = e: \]
\[f''\left( e \right) = \frac{- 3e - 2e \log e}{e^4} = \frac{- 5}{e^3} < 0\]
\[\text { So, x = e is a point of local maximum }. \]
\[\text { Thus, the local maximum value is given by}\]
\[f\left( e \right) = \frac{\log e}{e} = \frac{1}{e}\]
APPEARS IN
संबंधित प्रश्न
f(x) = - (x-1)2+2 on R ?
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
`f(x)=xsqrt(1-x), x<=1` .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
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?
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 tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
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.
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.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
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.
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.
Write the point where f(x) = x log, x attains minimum value.
The maximum value of x1/x, x > 0 is __________ .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
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 ________________ .
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:-
