Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 3x^4 − 4x^3 − 12x^2 + 1, [−2, 3]
Mathematics
annie1799
Question
Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 3x^4 − 4x^3 − 12x^2 + 1, [−2, 3]
1 Answer

1. User Answers LammettHash
Find the stationary points.
f(x) = 3x ⁴  4x ³  12x ² + 1
f '(x) = 12x ³  12 x ²  24x
Solve f '(x) = 0.
12x ³  12 x ²  24x = 12x (x ² + x  2) = 12x (x  1) (x + 2) = 0
→ x = 0, x = 1, x = 2
Check the value of f at the stationary points.
f (0) = 1
f (1) = 12
f (2) = 33
Check the value of f at the boundary of the domain.
f (3) = 28
(We've already checked f (2).)
Then over [2, 3], we have max(f ) = 33 and min(f ) = 12.