Let's analyze the graphs in the image to find the points where the functions do not have derivatives. These points typically occur at sharp corners, cusps, vertical tangents, or discontinuities. Here's the detailed breakdown for each problem:

Problem 35:

• The graph looks like a sharp "V" shape, indicating a cusp at $x = 0$.
• Therefore, the function does not have a derivative at $x = 0$.

Problem 36:

• This graph shows a discontinuity at $x = 0$, where the graph jumps.
• The function does not have a derivative at $x = 0$.

Problem 37:

• There appear to be three locations where the derivative does not exist:

  • A vertical tangent at $x = -2$.
  • A sharp corner at $x = 1$.
  • A discontinuity at $x = 4$.

• So, the function does not have a derivative at $x = -2$, $x = 1$, and $x = 4$.

Problem 38:

• This graph has multiple points where the derivative does not exist:

  • Sharp corners at $x = -2$, $x = 0$, and $x = 2$.

• Therefore, the function does not have a derivative at $x = -2$, $x = 0$, and $x = 2$.

Would you like more details on any of these? Here are 5 follow-up questions for further practice:

1. What is the formal definition of a derivative at a point?
2. How can you determine if a function is differentiable at a point by looking at its graph?
3. What is the significance of a vertical tangent in terms of differentiability?
4. Can a function have a continuous graph but still not be differentiable at certain points?
5. How does differentiability relate to continuity?

Tip:

If a function has a sharp corner or cusp, it is a clear indication that the function is not differentiable at that point, even if the function is continuous there.