Consider the function $g$ given by $g(x)=|x-7|+4$. (a) For what $x$-value(s) is the function not differentiable? (b) Evaluate $g^{\prime}(0), g^{\prime}(1), g^{\prime}(9)$, and $g^{\prime}(14)$.

Step 1 ：The function \(g(x) = |x-7| + 4\) is a piecewise function. It can be written as \(g(x) = -(x-7) + 4\) for \(x < 7\) and \(g(x) = (x-7) + 4\) for \(x \geq 7\).

Step 2 ：The derivative of \(g(x)\) is \(g'(x) = -1\) for \(x < 7\) and \(g'(x) = 1\) for \(x \geq 7\).

Step 3 ：The function \(g(x)\) is not differentiable at \(x = 7\) because the derivative is not continuous at this point.

Step 4 ：To evaluate \(g'(0)\), we use the derivative for \(x < 7\), which is \(g'(x) = -1\). So, \(g'(0) = -1\).

Step 5 ：To evaluate \(g'(1)\), we use the derivative for \(x < 7\), which is \(g'(x) = -1\). So, \(g'(1) = -1\).

Step 6 ：To evaluate \(g'(9)\), we use the derivative for \(x \geq 7\), which is \(g'(x) = 1\). So, \(g'(9) = 1\).

Step 7 ：To evaluate \(g'(14)\), we use the derivative for \(x \geq 7\), which is \(g'(x) = 1\). So, \(g'(14) = 1\).

Step 8 ：So, the function \(g(x) = |x-7| + 4\) is not differentiable at \(x = 7\), and \(g'(0) = g'(1) = -1\), \(g'(9) = g'(14) = 1\).