Advertisements
Advertisements
Question
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
Options
75
50
25
55
Advertisements
Solution
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
RELATED QUESTIONS
f(x) = 4x2 + 4 on R .
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
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]` .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
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 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?
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 \[\theta\] = \[\frac{\pi}{6}\] .
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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.
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 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.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at 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 ______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
