A function f(x) has a local minimum at x = c if:

Model Answer & Options

Source: Extra Practice

f'(c) = 0 and f''(c) < 0

f'(c) = 0 and f''(c) > 0

f'(c) > 0 and f''(c) = 0

f'(c) = 0 and f''(c) = 0

Explanation

According to the Second Derivative Test, for a function to have a local minimum at a point c, the first derivative must be zero (f'(c)=0, a stationary point) and the rate of change of the slope must be positive (f''(c)>0), meaning the curve is concave up. Option 1 describes a local maximum. Option 4 is inconclusive and requires further testing.

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