Advertisements
Advertisements
प्रश्न
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
पर्याय
75
50
25
55
Advertisements
उत्तर
75
\[\text { Given }: f\left( x \right) = x^2 + \frac{250}{x}\]
\[ \Rightarrow f'\left( x \right) = 2x - \frac{250}{x^2}\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 2x - \frac{250}{x^2} = 0\]
\[ \Rightarrow 2 x^3 - 250 = 0\]
\[ \Rightarrow x^3 = 125\]
\[ \Rightarrow x = 5\]
\[\text { Now,} \]
\[f''\left( x \right) = 2 + \frac{500}{x^3}\]
\[ \Rightarrow f''\left( 5 \right) = 2 + \frac{500}{5^3} = \frac{750}{125} = 6 > 0\]
\[\text { So, x = 5 is a local minima } . \]
\[ \therefore f' \left( x \right)_\min = 5^2 + \frac{250}{5} = \frac{375}{5} = 75\]
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x)=sin 2x+5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = (x \[-\] 5)4.
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) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
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.
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.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
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.
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 \[\theta\] = \[\frac{\pi}{6}\] .
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
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.
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
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.
Write the point where f(x) = x log, x attains minimum value.
Write the minimum value of f(x) = xx .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 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 minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
