Advertisements
Advertisements
प्रश्न
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]` .
Advertisements
उत्तर
\[\text { Given}: f\left( x \right) = 12 x^\frac{4}{3} - 6 x^\frac{1}{3} \]
\[ \Rightarrow f'\left( x \right) = 16 x^\frac{1}{3} - 2 x^\frac{- 2}{3} = \frac{2\left( 8x - 1 \right)}{x^\frac{2}{3}}\]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \frac{2\left( 8x - 1 \right)}{x^\frac{2}{3}} = 0\]
\[ \Rightarrow 8x - 1 = 0\]
\[ \Rightarrow x = \frac{1}{8}\]
\[\text { Thus, the critical points of f are } - 1, \frac{1}{8} \text { and }1 . \]
\[\text { Now }, \]
\[f\left( - 1 \right) = 12 \left( - 1 \right)^\frac{4}{3} - 6 \left( - 1 \right)^\frac{1}{3} = 18\]
\[f\left( \frac{1}{8} \right) = 12 \left( \frac{1}{8} \right)^\frac{4}{3} - 6 \left( \frac{1}{8} \right)^\frac{1}{3} = \frac{- 9}{4}\]
\[f\left( 1 \right) = 12 \left( 1 \right)^\frac{4}{3} - 6 \left( 1 \right)^\frac{1}{3} = 6\]
\[\text { Hence, the absolute maximum value when x = - 1 is 18 and the absolute minimum value when } x = \frac{1}{8}\text{ is }\frac{- 9}{4} . \]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x) = (x \[-\] 5)4.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
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.
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.
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 curve x2 = 8y which is nearest to the point (2, 4) ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
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 point where f(x) = x log, x attains minimum value.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
The maximum value of x1/x, x > 0 is __________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) 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 ______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
