PART A — Calculus. Show all working.

1. Differentiate f(x) = (x² · ln(3x)) / (x + 1). Simplify your result.
2. Evaluate the definite integral of (2x) / (x² + 1) from x = 0 to x = 2. Give the exact value, not a decimal.
3. Find all critical points of g(x) = x³ − 6x² + 9x + 2 and classify each as a local maximum, local minimum, or neither, using the second derivative test. State what the test cannot tell you and when.

PART B — Statistics.

A clinical trial reports: n = 240, mean reduction in systolic blood pressure of 4.1 mmHg, 95% confidence interval [0.3, 7.9], p = 0.038.

4. State precisely what the p-value does mean and what it does not mean. Name two common misinterpretations and say why each is wrong.
5. The confidence interval barely excludes zero. What does that tell you about the precision of the estimate, and why might this specific result fail to replicate?
6. The authors conclude the drug is "clinically significant." Explain the distinction between statistical significance and clinical significance, and state what additional information you would need to evaluate their claim.
7. Suppose the authors ran 20 outcome measures and reported this one. What is the name of that problem, what is the expected number of spurious "significant" results at α = 0.05, and what correction would you apply?
