Advertisements
Advertisements
Question
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
Options
3
0
4
2
Solution
2
\[\text { Given }: f\left( x \right) = 2 x^3 - 15 x^2 + 36x + 4\]
\[ \Rightarrow f'\left( x \right) = 6 x^2 - 30x + 36\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 6 x^2 - 30x + 36 = 0\]
\[ \Rightarrow x^2 - 5x + 6 = 0\]
\[ \Rightarrow \left( x - 2 \right)\left( x - 3 \right) = 0\]
\[ \Rightarrow x = 2, 3\]
\[\text { Now }, \]
\[f''\left( x \right) = 12x - 30\]
\[ \Rightarrow f''\left( 2 \right) = 24 - 30 = - 6 < 0\]
\[\text { So, x = 1 is a local maxima } . \]
\[\text { Also }, \]
\[f''\left( 3 \right) = 36 - 30 = 6 > 0\]
\[\text { So,x = 2 is a local maxima }.\]
APPEARS IN
RELATED QUESTIONS
f(x) = 4x2 + 4 on R .
f(x) = (x \[-\] 5)4.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = (x - 1) (x + 2)2.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second 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.
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.
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
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 point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
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.
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 point where f(x) = x log, x attains minimum value.
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x 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 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 _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
The minimum value of x loge x is equal to ____________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
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.
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Which of the following graph represents the extreme value:-
