Problem 1.

TRUE or FALSE? Prove or disprove the following statements (no points for correct answer without a valid proof):
a) (2 points) If $f \in C^{1}[a, b]$ and $\left|f^{\prime}(x)\right| \leq 1$ for all $x \in[a, b]$, then there can be at most one number $p \in[a, b]$ for which $f(p)=2 p$.
b) (2 points) If $f(x)$ is Lipschitz on $(-\infty, \infty)$, then so is $f(x)^{2}$ on $(-\infty, \infty)$.

Proof .

a) TRUE – Suppose there are two values $p, q$ with $p \neq q$ s.t. $f(p)=2 p$ and $f(q)=2 q$. Then by the MVT, there exists $\xi \in[a, b]$ s.t.
$$f^{\prime}(\xi)=\frac{f(p)-f(q)}{p-q}=\frac{2 p-2 q}{p-q}=2 .$$
But $\left|f^{\prime}(x)\right| \leq 1$ for $x \in[a, b]$, so by contradiction $p=q$.
b) FALSE – Consider e.g. $f(x)=x$, which is Lipschitz with $L=1$. But $f(x)=x^{2}$ is not, since $f^{\prime}(x)=2 x$ which is unbounded.

Problem 2.

a) (3 points) Show that the fixed point iteration
$$p_{n}=\frac{p_{n-1}^{2}+3}{5}, \quad n=1,2, \ldots$$
converges for any initial $p_{0} \in[0,1]$.
b) (1 point) Estimate how many iterations $n$ are required to obtain an absolute error $\left|p_{n}-p\right|$ less than $10^{-4}$ when $p_{0}=1$. No numerical value needed, just give an expression for $n$.

Proof .

a) Show $g(x) \in[0,1]$ for $x \in[0,1]$ :
\begin{aligned} &g(0)=3 / 5 \ &g(1)=4 / 5 \end{aligned}
increasing function
Show $\left|g^{\prime}(x)\right| \leq k<1$ :
$$g^{\prime}(x)=\frac{2 x}{5} \leq \frac{2}{5}=k<1$$
b)
$$10^{-4}=\left|p_{n}-p\right| \leq k^{n} \max {1,0}=\left(\frac{2}{5}\right)^{n} \Rightarrow n \approx \frac{-4}{\log _{10} \frac{2}{5}}$$

Problem 3.

Find the fourth degree polynomial $f(x)$ which satisfies the conditions
$$f(0)=1, f(1)=2, f^{\prime}(1)=\alpha, f(2)=2 \alpha, f^{\prime}(2)=2 \alpha$$

Proof .

$\Longrightarrow f(x)=1+x+(\alpha-1) x(x-1)-\frac{1}{2} x(x-1)^{2}+\left(\frac{9}{4}-\frac{\alpha}{2}\right) x(x-1)^{2}(x-2)$

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