Decision Tree for Application of Derivatives Problems:

Q: What type of problem?

Branch A: "Find intervals of increase/decrease"

1. Compute f'(x)
2. Factor f'(x) completely
3. Find zeros and undefined points of f'
4. Make sign chart
5. f' > 0 => increasing, f' < 0 => decreasing

Branch B: "Find local maxima/minima"

1. Find critical points (f'(c) = 0 or f'(c) DNE)
2. Is f''(c) easy to compute?
   • Yes and f''(c) != 0 => Use second derivative test
   • No or f''(c) = 0 => Use first derivative test (sign chart)
3. Report the point and the value f(c)

Branch C: "Find global max/min on [a, b]"

1. Find all critical points in (a, b)
2. Evaluate f at each critical point
3. Evaluate f at endpoints a and b
4. Compare all values: largest = global max, smallest = global min

Branch D: "Optimization word problem"

1. Draw diagram, identify objective and constraint
2. Express objective as f(x) using constraint
3. Determine domain of x
4. Find critical points via f'(x) = 0
5. Verify with second derivative test or endpoint check
6. Substitute back for the answer

Branch E: "Verify Rolle's/MVT and find c"

1. Check conditions: continuity on [a,b], differentiability on (a,b)
2. For Rolle's: also check f(a) = f(b)
3. Compute the required slope (0 for Rolle's, [f(b)-f(a)]/(b-a) for MVT)
4. Solve f'(c) = slope
5. Verify c is in (a, b)

Branch F: "Prove an inequality"

1. Define g(x) = LHS - RHS (or similar)
2. Show g(a) = 0 for some boundary point a
3. Show g'(x) > 0 (or < 0) on the relevant interval
4. Conclude g(x) > 0 (or < 0) by monotonicity

Branch G: "Find tangent/normal"

1. Find the point (x0, y0) on the curve
2. Compute slope m = f'(x0)
3. Tangent: y - y0 = m(x - x0)
4. Normal: y - y0 = (-1/m)(x - x0)
5. If m = 0: tangent is horizontal, normal is vertical
6. If m is undefined: tangent is vertical, normal is horizontal

Branch H: "Monotonicity with parameter"

1. Compute f'(x) in terms of the parameter
2. Set up condition f'(x) >= 0 (or <= 0) for all x
3. For quadratic f': discriminant <= 0 with correct leading coefficient sign
4. Solve for the parameter range