Advertisements
Advertisements
Question
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Advertisements
Solution
\[\text { We have }, \]
\[f\left( x \right) = ax + \frac{b}{x}\]
\[ \Rightarrow f'\left( x \right) = a - \frac{b}{x^2}\]
\[\text { For a local maxima or a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow a - \frac{b}{x^2} = 0\]
\[ \Rightarrow x^2 = \frac{b}{a}\]
\[ \Rightarrow x = \sqrt{\frac{b}{a}}, - \sqrt{\frac{b}{a}}\]
\[\text { But, }x > 0 \]
\[ \Rightarrow x = \sqrt{\frac{b}{a}}\]
\[\text { Now }, \]
\[f''\left( x \right) = \frac{2b}{x^3}\]
\[\text { At }x = \sqrt{\frac{b}{a}} \]
\[f''\left( \sqrt{\frac{b}{a}} \right) = \frac{2b}{\left( \sqrt{\frac{b}{a}} \right)^3} = \frac{2 a^\frac{3}{2}}{b^\frac{1}{2}} > 0 .....................\left[ \because a > 0 \text{ and }b > 0 \right]\]
\[\text { So }, x = \sqrt{\frac{b}{a}} \text { is a point of local minimum }. \]
\[\text { Hence, the least value is }\]
\[f\left( \sqrt{\frac{b}{a}} \right) = a\sqrt{\frac{b}{a}} + \frac{b}{\sqrt{\frac{b}{a}}} = \sqrt{ab} + \sqrt{ab} = 2\sqrt{ab}\]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = (x \[-\] 1) (x+2)2.
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
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) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
`f(x)=xsqrt(1-x), x<=1` .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
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 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.
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.
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
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.
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) = \[x + \frac{1}{x}, x > 0 .\]
Write the maximum value of f(x) = x1/x.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
For the function f(x) = \[x + \frac{1}{x}\]
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
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 _______________ .
The minimum value of x loge x is equal to ____________ .
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?
