1. Stating the problem: We want to minimize the function $$f(x_1,x_2) = (x_1 - \sqrt{5})^2 + (x_2 - \pi)^3 + 10$$ using the method of steepest ascent (gradient descent, since we minimize). 2. Calculate the gradient $\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}\right)$. $$\frac{\partial f}{\partial x_1} = 2(x_1 - \sqrt{5})$$ $$\frac{\partial f}{\partial x_2} = 3(x_2 - \pi)^2$$ 3. The steepest ascent method uses the negative gradient for minimization: $$x_{new} = x_{old} - \alpha \nabla f(x_{old})$$ where $\alpha > 0$ is the step size. 4. Using an initial guess $x_1^{(0)}, x_2^{(0)}$, update iteratively: $$x_1^{(k+1)} = x_1^{(k)} - \alpha \cdot 2(x_1^{(k)} - \sqrt{5})$$ $$x_2^{(k+1)} = x_2^{(k)} - \alpha \cdot 3(x_2^{(k)} - \pi)^2$$ 5. Repeat until convergence (when changes are very small). 6. The minimum occurs where the gradient is zero: Set derivatives equal to zero: $$2(x_1 - \sqrt{5}) = 0 \implies x_1 = \sqrt{5}$$ $$3(x_2 - \pi)^2 = 0 \implies x_2 = \pi$$ 7. Final minimum point: $$\boxed{(x_1, x_2) = (\sqrt{5}, \pi)}$$ At this point, $$f(\sqrt{5}, \pi) = (\sqrt{5} - \sqrt{5})^2 + (\pi - \pi)^3 + 10 = 0 + 0 + 10 = 10$$ This is the minimum value of the function.