Monotonicity Conditions:

• f'(x) > 0 on (a, b) => f strictly increasing on [a, b]
• f'(x) < 0 on (a, b) => f strictly decreasing on [a, b]
• f'(x) >= 0 on (a, b) with f'(x) = 0 only at isolated points => f strictly increasing
• f'(x) = 0 for all x in (a, b) => f constant on [a, b]

Critical Points:

• x = c is critical if f'(c) = 0 or f'(c) does not exist, provided f(c) is defined

First Derivative Test at x = c:

• f' changes + to - => local maximum at c
• f' changes - to + => local minimum at c
• f' does not change sign => no extremum (inflection point)

Second Derivative Test at x = c (where f'(c) = 0):

• f''(c) < 0 => local maximum
• f''(c) > 0 => local minimum
• f''(c) = 0 => inconclusive

Higher Order Derivative Test: If f'(c) = f''(c) = ... = f^(n-1)(c) = 0 and f^(n)(c) != 0:

• n even, f^(n)(c) < 0 => local maximum
• n even, f^(n)(c) > 0 => local minimum
• n odd => point of inflection (no extremum)

Global Extrema on [a, b]:

• Evaluate f at all critical points in (a, b) and at endpoints a, b
• Global max = largest value, Global min = smallest value

Rolle's Theorem: f continuous on [a, b], differentiable on (a, b), f(a) = f(b) => exists c in (a, b) with f'(c) = 0

Mean Value Theorem: f continuous on [a, b], differentiable on (a, b) => exists c in (a, b) with f'(c) = [f(b) - f(a)]/(b - a)

Lagrange's MVT Inequality Form: If m <= f'(x) <= M on (a, b), then m(b - a) <= f(b) - f(a) <= M(b - a)

Concavity:

• f''(x) > 0 => concave upward (cup)
• f''(x) < 0 => concave downward (cap)
• Point of inflection: f'' changes sign

Tangent and Normal:

• Tangent at (x0, y0): y - y0 = f'(x0)(x - x0)
• Normal at (x0, y0): y - y0 = [-1/f'(x0)](x - x0)
• Length of tangent = |y0|sqrt(1 + 1/[f'(x0)]^2)
• Length of normal = |y0|sqrt(1 + [f'(x0)]^2)
• Length of subtangent = |y0/f'(x0)|
• Length of subnormal = |y0 * f'(x0)|

Standard Optimization Results:

• Max area rectangle with perimeter P: square with side P/4
• Min perimeter rectangle with area A: square with side sqrt(A)
• Max volume open box from square sheet side a: x = a/6 (cut size)
• Max area rectangle under y = f(x): depends on function symmetry