If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

Geometrically: There's a point where tangent is parallel to secant line.

Special case of MVT: If f(a) = f(b) = 0, then there exists c ∈ (a, b) where f'(c) = 0.

If f is continuous on [a, b], then f attains both absolute maximum and absolute minimum on [a, b].

c is a critical number if:

1. f'(c) = 0, OR
2. f'(c) doesn't exist

Critical points are where extrema can occur.

• f'(x) > 0 on interval -> f is increasing
• f'(x) < 0 on interval -> f is decreasing
• f'(x) = 0 on interval -> f is constant

For critical number c:

1. Test sign of f' to left of c
2. Test sign of f' to right of c

• f' changes + to - -> local maximum at c
• f' changes - to + -> local minimum at c
• f' doesn't change sign -> no extremum

To find absolute extrema on [a, b]:

1. Find critical numbers in (a, b)
2. Evaluate f at critical numbers and endpoints
3. Largest value = absolute max, smallest = absolute min

Compare all candidate values to determine global extrema.

• Concave up: f''(x) > 0 (graph holds water)
• Concave down: f''(x) < 0 (graph spills water)

Use second derivative to determine concavity:

• f''(x) > 0 -> concave up
• f''(x) < 0 -> concave down
• f''(x) = 0 -> possible inflection point

For critical number c where f'(c) = 0:

• f''(c) > 0 -> local minimum at c
• f''(c) < 0 -> local maximum at c
• f''(c) = 0 -> test is inconclusive

• y-intercept: Set x = 0, solve for y
• x-intercepts: Set y = 0, solve for x

• Vertical: Where denominator = 0
• Horizontal: Limits at ±∞
• Slant: Polynomial division when degree difference = 1

Local max/min from first derivative test.

Use second derivative sign.

1. Domain: Find where f is defined
2. Intercepts: x- and y-intercepts
3. Symmetry: Even, odd, periodic
4. Asymptotes: Vertical, horizontal, slant
5. Intervals of increase/decrease: First derivative
6. Local extrema: Critical points
7. Concavity: Second derivative
8. Inflection points: Where concavity changes

• Where f increasing -> f' > 0
• Where f decreasing -> f' < 0
• Where f has max/min -> f' = 0
• Where f concave up -> f'' > 0 -> f' increasing
• Where f concave down -> f'' < 0 -> f' decreasing

1. Identify quantity to optimize
2. Draw diagram
3. Write constraint equation
4. Write optimization equation
5. Use constraint to reduce variables
6. Find critical numbers
7. Verify max/min

Apply derivative tests to optimization equation. Verify endpoint values if domain is restricted.

For curves defined implicitly:

• Find $\frac{dy}{dx}$ using implicit differentiation
• Analyze slope, tangents, extrema

Evaluate derivative at point to find tangent line slope.