**QUESTION**

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Can you answer all the questions mark in red including the last one please, thanks

Suppose you are an industrial designer working for an automotive company, you are given a car deisgn to work on: with a black flat body and a red parabolic fender. The design profile is specified as a piecewise equation \[ f(x)=\left\{\begin{array}{cc} 1 & 0 \leq x \leq 1 \\ a x^{2}+b x+c & 1<x \leq 2 \end{array}\right. \]

Your first job is to make the two pieces fit together. Have a go now with the $a, b$ and $c$ sliders. It's not an easy job! Mathematically, making the pieces fit together corresponds to the mathematical concept of $f$ being continuous at the point $x=1$ : \[ \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1) . \] Solving this equation gives you a condition on the coefficients $a, b$ and $c$, namely that Your second job is to make the two pieces fit together smoothly. This corresponds to the mathematical concept of $f$ being differentiable at the point $x=1$ : \[ \lim _{x \rightarrow 1^{-}} f^{\prime}(x)=\lim _{x \rightarrow 1^{+}} f^{\prime}(x) \] Solving this equation gives another condition on the coefficients $a, b$ and $c$, namely that So there is some freedom here. For example if you choose $a=-3$, then $b=$ and $c=$ Note: the Maple syntax for the condition $a+b c=0$ is $\mathrm{a}+\mathrm{b}^{\star} \mathrm{C}=0$.

Let's find values of $a$ and $b$ such that the piecewise defined function \[ f(x)=\left\{\begin{array}{ll} x^{3}+3 & x \leq 1 \\ a x+b & x>1 \end{array}\right. \] is differentiable. First we make the function continuous: - $\lim _{x \rightarrow 1^{-}} f(x)=$ 国这 - $\lim _{x \rightarrow 1^{+}} f(x)=$ 国国 - $f(1)=$ 因至. These three quantities are equal if $a$ and $b$ satisfy the condition: