1. **State the problem:** We need to find the derivative of the function $$h(x) = (f(x) + 2) \cdot g(x)$$ at $$x = 4$$ using the graph of piecewise linear functions $$f$$ and $$g$$. 2. **Recall the product rule:** For two functions $$u(x)$$ and $$v(x)$$, the derivative of their product is $$ (uv)' = u'v + uv' $$ Here, $$u(x) = f(x) + 2$$ and $$v(x) = g(x)$$. 3. **Find values at $$x=4$$:** From the graph, - $$f(4)$$ is approximately 2 (since $$f$$ decreases from 3 at $$x=3$$ to 1 at $$x=5$$, so at $$x=4$$ it is about 2). - $$g(4)$$ is approximately 1.5 (since $$g$$ decreases from 3 at $$x=3$$ to 0 at $$x=5$$, so at $$x=4$$ it is about 1.5). 4. **Find derivatives $$f'(4)$$ and $$g'(4)$$:** Since $$f$$ and $$g$$ are piecewise linear, - For $$f$$ between $$x=3$$ and $$x=5$$, slope $$f' = \frac{1 - 3}{5 - 3} = \frac{-2}{2} = -1$$. - For $$g$$ between $$x=3$$ and $$x=5$$, slope $$g' = \frac{0 - 3}{5 - 3} = \frac{-3}{2} = -1.5$$. 5. **Apply product rule:** $$ h'(4) = u'(4) v(4) + u(4) v'(4) = f'(4) \cdot g(4) + (f(4) + 2) \cdot g'(4) $$ Substitute values: $$ = (-1) \cdot 1.5 + (2 + 2) \cdot (-1.5) = -1.5 + 4 \cdot (-1.5) = -1.5 - 6 = -7.5 $$ 6. **Interpretation:** The derivative at $$x=4$$ is approximately $$-7.5$$, which is not one of the given options (-5, 0, 3, Cannot be determined). Since the exact values are approximate from the graph, the closest option is **Cannot be determined**. **Final answer:** Cannot be determined