Advertisements
Advertisements
प्रश्न
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
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+2 | on R .
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = x3\[-\] 6x2 + 9x + 15
`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) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
`f(x)=xsqrt(1-x), x<=1` .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum and minimum values of y = tan \[x - 2x\] .
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
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 absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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 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.
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
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 maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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.
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 the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
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 ___________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
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 _______________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
