First Derivative Test vs Second Derivative Test: When to Use Which

The first derivative test and second derivative test serve the same purpose — classifying critical points as maxima, minima, or neither — but they differ in methodology, reliability, and efficiency.

First Derivative Test:

• Method: Check sign of f'(x) on both sides of the critical point
• Reliability: Always conclusive — works in every situation
• Effort: Requires sign analysis in intervals, which may involve factoring or test-point evaluation
• Works when: f'(c) = 0 or f'(c) DNE; second derivative hard to compute; f''(c) = 0

Second Derivative Test:

• Method: Compute f''(c) at the critical point where f'(c) = 0
• Reliability: Inconclusive when f''(c) = 0
• Effort: Single computation — just evaluate f''(c)
• Works when: f''(c) is easy to compute and is nonzero

When the Second Derivative Test Fails: If f''(c) = 0, you have three options:

1. Fall back to the first derivative test (most common)
2. Use the higher-order derivative test: find the first nonzero derivative f^(n)(c). If n is even, it's an extremum (negative => max, positive => min). If n is odd, it's an inflection point.
3. Direct analysis of f(c + h) - f(c) for small h

JEE Strategy:

• For polynomial functions, the second derivative test is usually fastest — f'' is easy to compute
• For functions involving absolute values, GIF, or piecewise definitions, use the first derivative test (f'' may not exist)
• For functions like f(x) = $x^n$ * g(x), check if f''(c) = 0 before committing to the second derivative test
• In MCQs with "number of local maxima/minima," the first derivative test with a complete sign chart is more reliable
• For optimization word problems, the second derivative test is standard (just verify the sign of f'' at the critical point)

Efficiency Comparison: