A point on a curve is said to be an extremum if it is a local minimum or a…
2008
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve f(x) = 3x⁴ - 16x³ + 24x² + 37 is
- A.
0
- B.
1
- C.
2
- D.
3
Show answer & explanation
Correct answer: B
Given f(x) = 3x⁴ - 16x³ + 24x² + 37.
Differentiate:
f'(x) = 12x³ - 48x² + 48x = 12x(x² - 4x + 4) = 12x(x - 2)².
The critical points are x = 0 and x = 2.
For x < 0, f'(x) is negative. For x > 0, f'(x) is positive because (x - 2)² is always nonnegative and does not change sign around x = 2.
Thus f'(x) changes from negative to positive only at x = 0, giving one local minimum. At x = 2, the derivative is zero but does not change sign, so it is not an extremum.
Therefore, the number of distinct extrema is 1.