1. **Problem Statement:** We are given the function $$h(x) = x^4 - 3x^3 - 10x + 2$$ and need to find its first and second derivatives, critical points, and plot the function with these points. 2. **Formulas and Rules:** - The first derivative $$h'(x)$$ gives the slope of the function and helps find critical points where $$h'(x) = 0$$. - The second derivative $$h''(x)$$ indicates concavity and helps classify critical points. 3. **Calculate the first derivative:** $$h'(x) = \frac{d}{dx}(x^4 - 3x^3 - 10x + 2) = 4x^3 - 9x^2 - 10$$ 4. **Calculate the second derivative:** $$h''(x) = \frac{d}{dx}(4x^3 - 9x^2 - 10) = 12x^2 - 18x$$ 5. **Find critical points by solving $$h'(x) = 0$$:** $$4x^3 - 9x^2 - 10 = 0$$ This cubic equation can be solved using symbolic methods (e.g., sympy) or numerical approximation. 6. **Evaluate $$h(x)$$ at each critical point:** Substitute each critical point $$x = c$$ into $$h(x)$$ to find the corresponding $$y$$ values. 7. **Plotting:** - Plot $$h(x)$$ over a range (e.g., $$x \in [-5,5]$$). - Mark critical points on the graph. **Summary:** - $$h(x) = x^4 - 3x^3 - 10x + 2$$ - $$h'(x) = 4x^3 - 9x^2 - 10$$ - $$h''(x) = 12x^2 - 18x$$ - Critical points are solutions to $$4x^3 - 9x^2 - 10 = 0$$ - Plot shows function and critical points. This matches the Python code provided for symbolic differentiation, solving, and plotting.