Advertisements
Advertisements
Question
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Options
-128
-126
-120
none of these
Advertisements
Solution
`-128`
Given:-
`f(x)=2x^3-21x^2+36x-20`
`rArrf'(x)=6x^2-42x+36`
For a local maxima or a local minima, we must have
`f'(x)=0`
`rArr6x^2-42x+36=0`
`rArrx^2-7x+6=0`
`rArr(x-1)(x-6)=0`
`rArrx=1, 6`
Now,
`f''(x)=12x-42`
`rArrf''(1)=12-42=-30<0`
So, x = 1 is a local maxima.
Also,
`f''(6)=72-42=30>0`
So, x = 6 is a local minima.
The local minimum value is given by
`f(6)=2(6)^3-21(6)^2+36(6)-20=-128`
APPEARS IN
RELATED QUESTIONS
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x) = (x \[-\] 1) (x+2)2.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
`f(x) = 2/x - 2/x^2, x>0`
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
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 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 rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
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 ?
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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.
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.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
The maximum value of x1/x, x > 0 is __________ .
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 = _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
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?
