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Question

Piecewise functions can sometimes be differentiable at a point where two pieces meet. For the function $f$, use one-sided limits to find the values of $a$ and $b$ that make $f$ differentiable at $x=2$.

$f(x)=\left\{\begin{array}{l} a x^2+1, \quad \text { if } x<2 \\ -x^2+6 x+b, \quad \text { if } x \geq 2 \end{array}\right.$

Plot the graph using Boolean variables to restrict the domain, and sketch the result.

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