Step 1: Verify conditions

• Is f continuous on [a,b]? (Polynomials, exponentials, trig functions are continuous everywhere.)
• Is f differentiable on (a,b)? (Check for |x|, [x], cusps, vertical tangents in the interior.)
• For Rolle's: Is f(a) = f(b)?

Step 2: Compute the required value

• Rolle's: We need f'(c) = 0.
• LMVT: Compute [f(b)-f(a)]/(b-a).

Step 3: Find c

• Compute f'(x) in general.
• Set f'(c) = 0 (Rolle's) or f'(c) = [f(b)-f(a)]/(b-a) (LMVT).
• Solve for c and verify c is in (a,b).

Common pitfall: Finding c outside (a,b) — check the interval! If multiple solutions exist, pick the one in (a,b).