Advertisements
Advertisements
Question
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
Options
126
0
135
160
Solution
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is 0.
Explanation:
Let f(x) = x3 – 18x2 + 96x
So, f'(x) = 3x2 – 36x + 96
For local maxima and local minima f'(x) = 0
∴ 3x2 – 36x + 96 = 0
⇒ x2 – 12x + 32 = 0
⇒ x2 – 8x – 4x + 32 = 0
⇒ x(x – 8) – 4(x – 8) = 0
⇒ (x – 8)(x – 4) = 0
∴ x = 8, 4 ∈ [0, 9]
So, x = 4, 8 are the points of local maxima and local minima.
Now we will calculate the absolute maxima or absolute minima at x = 0, 4, 8, 9
∴ f(x)= x3 – 18x2 + 96x
`"f"(x)_(x = 0)` = 0 – 0 + 0 = 0
`"f"(x)_(x = 4)` = (4)3 – 18(4)2 + 96(4)
= 64 – 288 + 384
= 448 – 288
= 160
`"f"(x)_(x = 8)` = (8)3 – 18(8)2 + 96(8)
= 512 – 1152 + 768
= 1280 – 1152
= 128
`"f"(x)_(x = 9)` = (9)3 – 18(9)2 + 96(9)
= 729 – 1458 + 864
= 1593 – 1458
= 135
So, the absolute minimum value of f is 0 at x = 0
APPEARS IN
RELATED QUESTIONS
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Prove that the following function do not have maxima or minima:
g(x) = logx
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Show that among rectangles of given area, the square has least perimeter.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
The function f(x) = x log x is minimum at x = ______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
Divide 20 into two ports, so that their product is maximum.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The minimum value of the function f(x) = xlogx is ______.
Find the maximum and the minimum values of the function f(x) = x2ex.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
