Rolle's Theorem: f cont. [a,b], diff. (a,b), f(a)=f(b) => f'(c) = 0, c in (a,b)

LMVT: f cont. [a,b], diff. (a,b) => f'(c) = [f(b)-f(a)]/(b-a), c in (a,b)

Cauchy's MVT: f,g cont. [a,b], diff. (a,b), g'!=0 => f'$\frac{c}{g}$'(c) = [f(b)-f(a)]/[g(b)-g(a)]

Taylor (Lagrange remainder): f(x) = f(a) + f'(a)(x-a) + ... + f^(n)(c)(x-a)^n/n!

Darboux: f diff. on [a,b] => f' has IVP (takes every value between f'(a) and f'(b))

Root counting: n roots of f => at least n-1 roots of f'

LMVT inequality: m <= f' <= M on [a,b] => m(b-a) <= f(b)-f(a) <= M(b-a)

Lipschitz: |f'(x)| <= L for all x => |f(a)-f(b)| <= L|a-b|

Monotonicity: f' > 0 on (a,b) => f strictly increasing on [a,b]

Constant function: f' = 0 on (a,b) => f constant on [a,b]

Auxiliary functions:

• f'(c) + kf(c) = 0: use phi = e^(kx)f(x)
• nf(c) + cf'(c) = 0: use phi = $x^n$ f(x)
• f'(c) = f$\frac{c}{x}$: use phi = f$\frac{x}{x}$