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Consider the function given below and pick one or more CORRECT statement(s) from the following choices. f(x) = x³ - 15/2 x² + 18x + 20

1. f(x) has a local minimum at x = 3.
2. f(x) has a local maximum at x = 3.
3. f(x) has a local minimum at x = 2.
4. f(x) has a local maximum at x = 2.

Correct answer: f(x) has a local minimum at x = 3.

Solution

f'(x) = 3x^2 - 15x + 18 = 3(x-2)(x-3) gives critical points x=2 and x=3, and f''(x)=6x-15 gives f''(2)=-3<0 (local max at 2) and f''(3)=3>0 (local min at 3). So the correct statements are local min at x=3 and local max at x=2; the stored 'local maximum at x=3' is wrong.

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