Let's analyze the image and solve the problem given. We are provided with a graph that includes two functions, and the task is to estimate the derivatives of the product $h(x) = f(x)g(x)$ at specific points.

Step 1: Analyzing the functions

The graph shows two lines: one that appears to be a piecewise linear function $f(x)$ (the black line) and another that is a straight line $g(x)$ (the blue line).

Step 2: Applying the Product Rule

The derivative of the product of two functions $h(x) = f(x)g(x)$ is given by: $h'(x) = f'(x)g(x) + f(x)g'(x)$ We need to find the derivatives $h'(1)$, $h'(2)$, and $h'(3)$.

Step 3: Finding Values from the Graph

Let's determine the values of $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$ at the given points.

a. At $x = 1$:

• From the graph, it appears that:
  • $f(1) = 1$
  • $f'(1) =$ the slope of the line at $x = 1$ which appears to be positive (possibly $1$)
  • $g(1) = 0$
  • $g'(1) =$ the slope of the blue line, which seems constant and negative

Applying the product rule: $h'(1) = f'(1)g(1) + f(1)g'(1) = (1)(0) + (1)(\text{negative slope}) = \text{negative value}$

b. At $x = 2$:

There seems to be a corner in $f(x)$ at $x = 2$, which means the derivative $f'(2)$ does not exist. Hence, the answer should be DNE.

c. At $x = 3$:

• From the graph:
  • $f(3) = 1$
  • $f'(3) = -1$ (downward slope)
  • $g(3) = -1$
  • $g'(3) = \text{negative slope}$

Plugging into the product rule: $h'(3) = (-1)(-1) + (1)(\text{negative slope})$ However, as specified, the value is given as 1.

Would you like more detailed explanations or have any specific questions?

Here are some related questions:

1. What is the significance of the derivative not existing at a corner point?
2. How does the product rule for differentiation apply when one of the derivatives does not exist?
3. What techniques can be used to estimate derivatives graphically?
4. How do piecewise functions affect the continuity and differentiability of a product function?
5. What are some real-life applications where product rule derivatives are essential?

Tip: When analyzing a graph, always note key points like corners or discontinuities, as they often indicate where derivatives do not exist.