Advertisements
Advertisements
Question
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Advertisements
Solution
\[\text { Given }: \hspace{0.167em} f\left( x \right) = 3 x^4 - 8 x^3 + 12 x^2 - 48x + 25\]
\[ \Rightarrow f'\left( x \right) = 12 x^3 - 24 x^2 + 24x - 48\]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 12 x^3 - 24 x^2 + 24x - 48 = 0\]
\[ \Rightarrow x^3 - 2 x^2 + 2x - 4 = 0\]
\[ \Rightarrow x^2 \left( x - 2 \right) + 2\left( x - 2 \right) = 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x^2 + 2 \right) = 0\]
\[ \Rightarrow x - 2 = 0 or \left( x^2 + 2 \right) = 0 \]
\[ \Rightarrow x = 2 \]
\[\text { No real root exists for } \left( x^2 + 2 \right) = 0 . \]
\[\text { Thus, the critical points of f are 0, 2 and 3 } . \]
\[\text { Now }, \]
\[f\left( 0 \right) = 3 \left( 0 \right)^4 - 8 \left( 0 \right)^3 + 12 \left( 0 \right)^2 - 48\left( 0 \right) + 25 = 25\]
\[f\left( 2 \right) = 3 \left( 2 \right)^4 - 8 \left( 2 \right)^3 + 12 \left( 2 \right)^2 - 48\left( 2 \right) + 25 = - 39\]
\[f\left( 3 \right) = 3 \left( 3 \right)^4 - 8 \left( 3 \right)^3 + 12 \left( 3 \right)^2 - 48\left( 3 \right) + 25 = 16\]
\[\text { Hence, the absolute maximum value when x = 0 is 25 and the absolute minimum value when x = 2 is - 39 } . \]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
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]` .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
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.
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.
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 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) = \[x + \frac{1}{x}, x > 0 .\]
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
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 _______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
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 ______________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Which of the following graph represents the extreme value:-
