Q. 535.0( 2 Votes )

# The smallest value of the polynomial x^{3} – 18x^{2} + 96x in [0, 9] is

A. 126

B. 0

C. 135

D. 160

Answer :

Let f(x)= x^{3} – 18x^{2} + 96x

Applying the first derivative we get

Applying the sum rule of differentiation, so we get

Applying the derivative,

⇒ f' (x)=3x^{2}-36x+96

Putting f’(x)=0,we get critical points

3x^{2}-36x+96=0

⇒ 3(x^{2}-12x+32)=0

⇒ x^{2}-12x+32=0

Splitting the middle term, we get

⇒ x^{2}-8x-4x+32=0

⇒ x(x-8)-4(x-8)=0

⇒ (x-8)(x-4)=0

⇒ x-8=0 or x-4=0

⇒ x=8 or x=4

⇒ x∈[0,9]

Now we will find the value of f(x) at x=0, 4, 8, 9

f(x)= x^{3} – 18x^{2} + 96x

f(0)= 0^{3} – 18(0)^{2} + 96(0)=0

f(4)= 4^{3} – 18(4)^{2} + 96(4)=64-288+384=160

f(8)= 8^{3} – 18(8)^{2} + 96(8)=512-1152+768=128

f(9)= 9^{3} – 18(9)^{2} + 96(9)=729-1458+864=135

Hence we find that the absolute minimum value of f(x) in [0,9] is 0 at x=0.

So the correct option is option B.

Rate this question :

A metal box with a square base and vertical sides is to contain 1024 cm^{3}. The material for the top and bottom costs ` 5 per cm^{2} and the material for the sides costs ` 2.50 per cm^{2}. Find the least cost of the box.

Show that among all positive numbers x and y with x^{2} + y^{2} = r^{2}, the sum x + y is largest when x = y = .

Find the local maxima and local minima, of the function f(x) sin x – cos x, 0< x < 2π .Also, find the local maximum and local minimum values.

Mathematics - Board PapersProve that the semi-vertical angle of the right circular cone of given volume and least curved surface area is cot-1√2.

Mathematics - Board PapersProve that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is

Mathematics - Board Papers