Answer to Question #332622 in Calculus for Olga

Question

Y =  1 /3  x^3  –  x^2 – 3x  +  2

Is

 

a. (−1, 11 / 3) is a maximum and (3,-7) is a minimum

b. (−1,0) is a maximum and (3,0) is a minimum

c. (3,0) is a maximum and (-1,0) is a minimum

d. (3,−7) is a maximum and (−1,11 / 3) is a minimum

Answer 1

Solution

For f(x) = (1 /3) x³Â â€“ x² – 3x + 2 points of extremum are the solutions of equation

f’(x) = x² – 2x – 3  = 0   =>   x₁ = -1, x₂ = 3

Let’s calculate at these points the second derivative f’’(x) = 2x – 2.

f’’(x₁) = -4 , f’’(x₂) = 4

So point x₁ = -1 is point of maximum where f(x₁) = f(-1) = 11/3 and x₂ = 3 is point of minimum where f(x₂) = f(3) = -7.

Therefore the correct answer is

a. (−1, 11 / 3) is a maximum and (3,-7) is a minimum