Rolle's Theorem — verify:

1. Continuous on [a,b]
2. Differentiable on (a,b)
3. f(a) = f(b) Then f'(c) = 0 for some c in (a,b)

LMVT — apply when:

1. Need to relate function values to derivatives
2. Proving inequalities
3. Bounding function growth f'(c) = [f(b)-f(a)]/(b-a)

Root counting checklist:

1. Find critical points (f' = 0)
2. Evaluate f at critical points and boundaries
3. Count sign changes (IVT for lower bound)
4. Use Rolle's chain for upper bound
5. Combine for exact count

Auxiliary function patterns:

• f' + kf = 0: phi = e^(kx)f
• f'g + fg' = 0: phi = fg
• f' = 2c: phi = f - $x^2$

Common mistakes:

• Forgetting to check differentiability in the open interval
• Applying Rolle's when f(a) != f(b)
• Confusing "at most" and "at least" in root counting
• Applying LMVT to non-differentiable functions (e.g., |x-a| at x = a)

Key inequalities from LMVT:

• |sin a - sin b| <= |a - b|
• |tan a - tan b| >= |a - b| (for a,b in (-pi/2, pi/2))
• $e^x$ > 1 + x for x != 0
• $\frac{x}{1+x}$ < ln(1+x) < x for x > 0