Advertisements
Advertisements
प्रश्न
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Advertisements
उत्तर
\[\text { Given }: \hspace{0.167em} f\left( x \right) = 3 x^4 - 8 x^3 + 12 x^2 - 48x + 25\]
\[ \Rightarrow f'\left( x \right) = 12 x^3 - 24 x^2 + 24x - 48\]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 12 x^3 - 24 x^2 + 24x - 48 = 0\]
\[ \Rightarrow x^3 - 2 x^2 + 2x - 4 = 0\]
\[ \Rightarrow x^2 \left( x - 2 \right) + 2\left( x - 2 \right) = 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x^2 + 2 \right) = 0\]
\[ \Rightarrow x - 2 = 0 or \left( x^2 + 2 \right) = 0 \]
\[ \Rightarrow x = 2 \]
\[\text { No real root exists for } \left( x^2 + 2 \right) = 0 . \]
\[\text { Thus, the critical points of f are 0, 2 and 3 } . \]
\[\text { Now }, \]
\[f\left( 0 \right) = 3 \left( 0 \right)^4 - 8 \left( 0 \right)^3 + 12 \left( 0 \right)^2 - 48\left( 0 \right) + 25 = 25\]
\[f\left( 2 \right) = 3 \left( 2 \right)^4 - 8 \left( 2 \right)^3 + 12 \left( 2 \right)^2 - 48\left( 2 \right) + 25 = - 39\]
\[f\left( 3 \right) = 3 \left( 3 \right)^4 - 8 \left( 3 \right)^3 + 12 \left( 3 \right)^2 - 48\left( 3 \right) + 25 = 16\]
\[\text { Hence, the absolute maximum value when x = 0 is 25 and the absolute minimum value when x = 2 is - 39 } . \]
APPEARS IN
संबंधित प्रश्न
f(x)=sin 2x+5 on R .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other 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 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?
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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
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.
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.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
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 minimum value of \[\frac{x}{\log_e x}\] is _____________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
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 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 = ______________ .
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 __________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
