Questions: 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'(0), g'(1), g'(8), and g'(10). (a) The function is not differentiable at x = .

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'(0), g'(1), g'(8), and g'(10).
(a) The function is not differentiable at x = .

Transcript text: Consider the function $g$ given by $g(x)=|x-7|+4$. (a) For what $x$-value(s) is the function not differentiable? (b) Evaluate $g^{\prime}(0), g^{\prime}(1), g^{\prime}(8)$, and $g^{\prime}(10)$. (a) The function is not differentiable at $\mathrm{x}=$ $\square$

Solution

Solution Steps

Step 1: Identify Non-Differentiable Point

The function $ g(x) = |x - 7| + 4 $ is not differentiable at the point where the absolute value expression creates a cusp. This occurs when the argument of the absolute value is zero:

\[ x - 7 = 0 \implies x = 7 \]

Thus, the function is not differentiable at $ x = 7 $.

Step 2: Evaluate Derivative at Specific Points

To evaluate the derivative $ g'(x) $ at the specified points, we consider the behavior of the function on either side of the cusp at $ x = 7 $.

For $ x < 7 $: \[ g(x) = -(x - 7) + 4 = -x + 11 \implies g'(x) = -1 \]
For $ x > 7 $: \[ g(x) = (x - 7) + 4 = x - 3 \implies g'(x) = 1 \]

Now we evaluate $ g'(x) $ at the given points:

$ g'(0) = -1 $ (since $ 0 < 7 $)
$ g'(1) = -1 $ (since $ 1 < 7 $)
$ g'(8) = 1 $ (since $ 8 > 7 $)
$ g'(10) = 1 $ (since $ 10 > 7 $)

Final Answer

The answers to the sub-questions are:

(a) The function is not differentiable at $ x = 7 $.
(b) The values of the derivatives are:
- $ g'(0) = -1 $
- $ g'(1) = -1 $
- $ g'(8) = 1 $
- $ g'(10) = 1 $

Thus, the final answers are: \[ \boxed{x = 7} \] \[ \boxed{g'(0) = -1} \] \[ \boxed{g'(1) = -1} \] \[ \boxed{g'(8) = 1} \] \[ \boxed{g'(10) = 1} \]

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