Advertisements
Advertisements
प्रश्न
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
Advertisements
उत्तर
\[ f\left( x \right) = x^3 - 2a x^2 + a^2 x\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 4ax + a^2 \]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 4ax + a^2 = 0\]
\[ \Rightarrow 3 x^2 - 3ax - ax + a^2 = 0\]
\[ \Rightarrow 3x\left( x - a \right) - a\left( x - a \right) = 0\]
\[ \Rightarrow \left( 3x - a \right)\left( x - a \right) = 0\]
\[ \Rightarrow x = a \text { and } \frac{a}{3}\]
\[\text { Thus, x = a and x } = \frac{a}{3}\text { are the possible points of local maxima or local minima }. \]
\[\text { Now,} \]
\[f''\left( x \right) = 6x - 4a\]
\[\text { At } x = a: \]
\[f''\left( a \right) = 6\left( a \right) - 4a = 2a > 0\]
\[\text { So, x = a is the point of local minimum }. \]
\[\text { The local minimum value is given by }\]
\[f\left( a \right) = a^3 - 2a \left( a \right)^2 + a^2 \left( a \right) = 0\]
\[\text { At }x = \frac{a}{3}: \]
\[ f''\left( \frac{a}{3} \right) = 6\left( \frac{a}{3} \right) - 4a = - 2a < 0\]
\[\text { So }, x = \frac{a}{3} \text { is the point of local maximum }. \]
\[\text { The local maximum value is given by }\]
\[f\left( \frac{a}{3} \right) = \left( \frac{a}{3} \right)^3 - 2a \left( \frac{a}{3} \right)^2 + a^2 \left( \frac{a}{3} \right) = \frac{a^3}{27} - \frac{2 a^3}{9} + \frac{a^3}{3} = \frac{4 a^3}{27}\]
APPEARS IN
संबंधित प्रश्न
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = (x - 1) (x + 2)2.
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
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 ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
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]` .
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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
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 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 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 \[\frac{2R}{\sqrt{3}} .\]
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
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 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.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The maximum value of x1/x, x > 0 is __________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 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 ______________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] 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 = ______________ .
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.
