To tackle this problem, let's go through each part:

Part (a)

1. Determine Differentiability of $f(x) = |x^2 - 9|$: The function $f(x) = |x^2 - 9|$ involves an absolute value, which can cause issues with differentiability at points where $x^2 - 9 = 0$.

   Solving $x^2 - 9 = 0$: $x^2 = 9 \implies x = \pm 3$ So, the points $x = 3$ and $x = -3$ may be points where $f(x)$ is not differentiable, because the absolute value function $|u|$ is not differentiable at $u = 0$.

2. Express $f(x)$ Piecewise: Rewrite $f(x)$ without the absolute value by considering cases for $x^2 - 9$:

   • When $x > 3$ or $x < -3$, $x^2 - 9 > 0$, so $f(x) = x^2 - 9$.
   • When $-3 \leq x \leq 3$, $x^2 - 9 \leq 0$, so $f(x) = -(x^2 - 9) = 9 - x^2$.

   Thus:

   x^2 - 9 & \text{if } |x| > 3, \\ 9 - x^2 & \text{if } |x| \leq 3. \end{cases}$$

3. Find $f'(x)$: Now, we can differentiate each piece separately:

   • For $x > 3$ or $x < -3$: $f(x) = x^2 - 9 \Rightarrow f'(x) = 2x$.
   • For $-3 < x < 3$: $f(x) = 9 - x^2 \Rightarrow f'(x) = -2x$.

   At $x = \pm 3$, there is a sharp corner due to the absolute value, so $f(x)$ is not differentiable at $x = 3$ and $x = -3$.

   Therefore:

   2x & \text{if } |x| > 3, \\ -2x & \text{if } |x| < 3. \end{cases}$$

Part (b)

To sketch the graphs of $f(x)$ and $f'(x)$:

1. Graph of $f(x)$:

   • For $|x| > 3$, plot $f(x) = x^2 - 9$, a parabola opening upwards shifted down by 9.
   • For $|x| \leq 3$, plot $f(x) = 9 - x^2$, a parabola opening downwards capped at $x = \pm 3$.

2. Graph of $f'(x)$:

   • For $|x| > 3$, $f'(x) = 2x$, a linear function with a positive slope.
   • For $|x| < 3$, $f'(x) = -2x$, a linear function with a negative slope.
   • $f'(x)$ has a discontinuity at $x = \pm 3$.

Would you like further details on any part, or assistance with the sketches?

Relative Questions

1. How does the absolute value affect differentiability in this function?
2. Why is $f(x)$ not differentiable at $x = \pm 3$?
3. What would $f'(x)$ look like if there were no absolute value in $f(x)$?
4. How can we confirm differentiability of piecewise functions in general?
5. How does the derivative $f'(x)$ represent changes in the graph of $f(x)$?

Tip

When dealing with absolute value functions, remember to analyze the differentiability at points where the inside expression equals zero, as these points often lead to non-differentiable behavior.