When asked to verify MVT:

1. Check continuity on [a,b]
2. Check differentiability on (a,b)
3. For Rolle's: verify f(a) = f(b)
4. Compute the RHS: 0 (Rolle's) or [f(b)-f(a)]/(b-a) (LMVT)
5. Solve f'(c) = RHS and verify c is in (a,b)

When asked to prove an inequality:

1. Identify the function and interval
2. Apply LMVT: f(b)-f(a) = f'(c)(b-a)
3. Bound f'(c) using a < c < b
4. Substitute bounds to get the inequality

When asked about number of roots:

1. Compute f' and find its zeros
2. Use Rolle's chain: k zeros of f' => at most k+1 zeros of f
3. Use IVT (sign changes) to establish minimum number of roots
4. Combine for exact count

When asked to prove f'(c) = expression:

1. Try to find phi(x) whose derivative involves the expression
2. Common: phi = e^(kx)*f, phi = $x^n$*f, phi = $\frac{f}{g}$
3. Verify phi(a) = phi(b), apply Rolle's