1. The problem is to minimize the function $$z = (x_1 - \sqrt{5})^2 + (x_2 - \pi)^2 + 10$$ using the method of steepest ascent. 2. We convert the minimization to a maximization by considering $$z = -(x_1 - \sqrt{5})^2 - (x_2 - \pi)^2 - 10$$. 3. The given gradient is $$\nabla f = \begin{bmatrix} -2(x_1 - \sqrt{5}) \\ -2(x_2 - \pi) \end{bmatrix}$$. 4. The initial guess is $$X_0 = \begin{bmatrix} 6.597 \\ 5.891 \end{bmatrix}$$ with corresponding $$z = -36.58$$. 5. For the first iteration, the update step is $$X_0 + \lambda \nabla f|_{X_0} = \begin{bmatrix} 6.597 - 8.722\lambda \\ 5.891 - 5.499\lambda \end{bmatrix}$$. 6. Objective function in terms of $$\lambda$$ is: $$f(X_0 + \lambda \nabla f|_{X_0}) = -106.3\lambda^2 + 106.3\lambda -36.58$$. 7. Maximizing this quadratic gives $$\lambda^* = 0.5$$. 8. Therefore, $$X_1 = X_0 + 0.5 \nabla f|_{X_0} = \begin{bmatrix} 6.597 - 8.722 \times 0.5 \\ 5.891 - 5.499 \times 0.5 \end{bmatrix} = \begin{bmatrix} 2.236 \\ 3.142 \end{bmatrix}$$, with $$f(X_1) = -10.00$$. 9. For the second iteration, $$X_1 + \lambda \nabla f|_{X_1} = \begin{bmatrix} 2.236 + 0.0001\lambda \\ 3.142 - 0.0008\lambda \end{bmatrix}$$. 10. The function in terms of $$\lambda$$ is approximately $$f(X_1 + \lambda \nabla f|_{X_1}) = - (6.500\lambda^2 - 6.382\lambda + 10) \times 10^{-7}$$. 11. Maximizing gives $$\lambda_1^* = 0.4909$$. 12. The updated point is $$X_2 = X_1 + 0.4909 \nabla f|_{X_1} = \begin{bmatrix} 2.236 \\ 3.142 \end{bmatrix}$$. 13. Since $$X_2$$ equals $$X_1$$ to four significant figures, the solution is $$X^* = \begin{bmatrix} 2.236 \\ 3.142 \end{bmatrix}$$, with minimum value of original function $$z^* = 10.00$$. Thus, the method of steepest ascent was used to find the minimum of the given function starting from a sampled point, converging after two iterations to final answer $X^* = [2.236, 3.142]^T$ with $z^* = 10.00$.