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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.

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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.

We consider the graph of g(t) shown, and answer questions related to differentiability, zero derivatives, and slope comparisons.

Analyze the graph of g(t) to determine points of non-differentiability, critical points where g'(t) = 0, compare the slopes at t = 6 and t = 8, and estimate g'(9) from the graph. This calculus problem helps students practice concepts related to derivatives and graph interpretation.

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