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) = | sin 4x+3 | on R ?
f(x) = x3 (x \[-\] 1)2 .
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin 2x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x3\[-\] 6x2 + 9x + 15
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
`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 ?
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
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 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.
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
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?
Write sufficient conditions for a point x = c to be a point of local maximum.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the minimum value of f(x) = xx .
Write the maximum value of f(x) = x1/x.
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
If x+y=8, then the maximum value of xy is ____________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
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 function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
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.
