Increasing/decreasing intervals and local extrema
Use the first derivative of a cubic to find and classify its local extrema. Unlimited questions, five difficulty levels, and a full worked solution every time — free.
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Find the x-coordinate of the local minimum of f.
f(x) = 2x3 + 12x2
Answer: 0
See one solved, step by step
Find the x-coordinate of the local maximum of f.
f(x) = 4x3 + 6x2
📘 Worked solution
1f'(x) = 12x2 + 12xDifferentiate term by term using the power rule.
212x2 + 12x = 12x(x - (−1))Factor out the common x — the critical points are where f'(x) = 0.
3Critical points: x = 0 and x = −1Both factors give a zero of f'(x).
4f''(x) = 24x + 12Differentiate f'(x) again to get the concavity test.
5f''(0) = 12 > 0 (local min), f''(−1) = −12 < 0 (local max)f'' < 0 means the curve is concave down (a local max); f'' > 0 means concave up (a local min).
6Local maximum at x = −1Read off the critical point with the matching concavity.
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