Let's go through the problem together based on the graph of $g(t)$ and the questions:

(a) At which $t$-values (if any) is $g(t)$ not differentiable? Explain.

A function is not differentiable at points where:

• There is a sharp corner or cusp.
• The function is discontinuous.
• The slope is vertical (infinite).

From the graph, look for any sharp corners or discontinuities in $g(t)$.

(b) For which $t$-values (if any) do we have $g'(t) = 0$? Explain.

To find where $g'(t) = 0$, look for points where the tangent line to the graph is horizontal. These points correspond to local maxima, minima, or inflection points.

(c) Which value is greater, $g'(6)$ or $g'(8)$? How do you know?

We are comparing the slopes of the tangent lines at $t = 6$ and $t = 8$. Determine which slope is steeper or has a larger magnitude based on the graph's steepness at these points.

(d) Use the graph to estimate $g'(9)$.

To estimate $g'(9)$, look at the slope of the tangent line at $t = 9$, and approximate its value.

Would you like me to walk you through the steps for each part of the solution in more detail, or clarify any specific part?

Here are 5 related questions:

1. What are the conditions for a function to be differentiable at a point?
2. How do critical points relate to $g'(t) = 0$?
3. How can you tell if the slope of a function is positive, negative, or zero just by looking at the graph?
4. What are the geometric interpretations of the derivative of a function?
5. Why might $g'(6)$ be greater or smaller than $g'(8)$ based on graph behavior?

Tip: When analyzing a graph for differentiability, always check for sharp turns, cusps, and vertical tangents.