Homework 16: The Mean Value Theorem

Instructions. Assignments should be stapled and written clearly and legibly. Problem 6 is optional.

1. Secion 6.2, #16.
2. Find a twice differentiable function \(f(x)\) such that \(f'(1)=-1\), \(f'(4)=7\), and \(f''(x)>3\) for all \(x\), or prove that such a function cannot exist.
3. Suppose that \(f:[a,b]\to \mathbb{R}\) has continuous derivatives \(f'\) and \(f''\), and assume there exists \(c\in (a,b)\) such that \(f(a)=f(c)=f(b)\). Prove that there exists \(d\) in \((a,b)\) such that \(f''(d)=0\). Hint. First prove that there exist \(x_1\) and \(x_2\) such that \(a \lt x_1 \lt x_2 \lt b\) and \(f'(x_1)=f'(x_2)=0\).
4. Recall that a number \(c\) is called a fixed point of a function \(f\) if \(f(c)=c\). Prove that if \(f\) is a differentiable function and \(f'(x)\neq 1\) for all real numbers \(x\), then \(f\) has at most one fixed point.
5. Suppose that \(f, g\) are differentiable functions and \(f'=g\) and \(g'=-f\). Prove that \(h(x)=(f(x))^2+(g(x))^2\) is a constant function. (Hint: Use the chain rule.)
6. Let \(h\) be Thomae's function: \begin{equation} h(x)= \begin{cases} 1 & \text{if }x=0\\ 1/n & \text{if }x=m/n\text{ in lowest terms with }n>0\\ 0 & \text{if }x\notin \mathbb{Q} \end{cases} \end{equation} Prove that \(h\) is continuous at every \(x\notin \mathbb{Q}\) and discontinuous at every \(x\in \mathbb{Q}\).