Advertisements
Advertisements
Question
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
Options
3
0
4
2
Advertisements
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) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x) = 2/x - 2/x^2, x>0`
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Find the maximum and minimum values of y = tan \[x - 2x\] .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
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 tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
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 ?
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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.
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.
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
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 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.
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write the maximum value of f(x) = x1/x.
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
If x lies in the interval [0,1], then the least value of x2 + x + 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) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Which of the following graph represents the extreme value:-
