At critical point c where f'(c) = 0:

• f''(c) < 0 => local maximum (function is concave down at c)
• f''(c) > 0 => local minimum (function is concave up at c)
• f''(c) = 0 => INCONCLUSIVE — must use first derivative test or higher derivatives

When f''(c) = 0 (inconclusive): Use the higher-order derivative test: Find the first non-zero derivative f^(n)(c) where n >= 2.

• If n is even and f^(n)(c) < 0: local max
• If n is even and f^(n)(c) > 0: local min
• If n is odd: point of inflection (no extremum)

Example where second test fails: f(x) = $x^4$. f'(0) = 0, f''(0) = 0 (inconclusive). f'''(0) = 0, f''''(0) = 24 > 0. Since n = 4 is even and positive, local minimum at x = 0.

JEE Tip: If the second derivative is easy to compute and non-zero, use it. If f''(c) = 0 or f'' is complicated, default to the first derivative test (sign chart).