Homework 7 was due on Thursday 10th October 2013 and consisted of:
I marked 4.2/22(a) and 4.2/24, each out of 4; two free marks were given for submitting the homework, provided Q44 was completed.
Section 4.1 Q44. The critical numbers of a function are the numbers in the domain of the function where its derivative is either zero or undefined. Most people did this question correctly, but a few people indicated that $x=0$ is a critical number, which is incorrect because $0$ is not in the domain $x^{-2}\ln x$.
Section 4.2 Q22(a). The trick here was to notice that if $f$ has two roots then there are points $a$ and $b$, with $a \lt b$, say, such that $f(a)=f(b)=0$. The assertion that $f'$ has a root is then precisely the conclusion of Rolle's theorem.
Section 4.2 Q24. Most people did this correctly. Using the mean value theorem we know there is a value $c$ with $2 \lt x \lt 8$ such that $$f'(c) = \frac{f(8)-f(2)}{8-2}$$ Applying the inequality $3 \le f'(c) \le 5$ and rearranging gives the desired result.