Advertisements
Advertisements
Question
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
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) = x3 \[-\] 3x .
f(x) = x3 (x \[-\] 1)2 .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = (x - 1) (x + 2)2.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
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.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
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.
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
Write sufficient conditions for a point x = c to be a point of local maximum.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
The maximum value of x1/x, x > 0 is __________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 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 function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
