The auxiliary function technique is the most powerful problem-solving tool for MVT questions. The idea: to prove some equation involving f, f', and x holds at some point c, find a function phi whose derivative involves the equation, then apply Rolle's to phi.

Key patterns:

Pattern 1: Prove f'(c) + kf(c) = 0. Use phi(x) = e^(kx)f(x). Then phi' = e^(kx)(f' + kf). If phi(a) = phi(b), Rolle's gives the result.

Pattern 2: Prove nf(c) + cf'(c) = 0. Use phi(x) = $x^n$ * f(x). Then phi' = x^(n-1)(nf + xf'). If boundaries work, apply Rolle's.

Pattern 3: Prove f'(c) = g'(c). Use phi(x) = f(x) - g(x). If f(a)=g(a) and f(b)=g(b), phi(a)=phi(b)=0, Rolle's applies.

Pattern 4: Prove f'(c) = 2c (or similar polynomial in c). Use phi(x) = f(x) - $x^2$ (or f minus the antiderivative of the desired expression). Match endpoints.

How to discover the right auxiliary function: Look at the target equation and try to recognize it as d/dx[something]. If the equation is f'g + fg' = 0, that's d/dx[fg] = 0, so phi = fg. If it's f' + kf = 0, that's (d/dx[e^(kx)f])/e^(kx) = 0, so phi = e^(kx)f.