A function \( f \) and a point \( P \) are given. Let \( \theta \) correspond to the direction of the directional derivative. Complete parts a. through \( e \). \( f(x, y)=\sqrt{3+2 x^{2}+2 y^{2}}, P(\sqrt{3},-1) \) a. Find the gradient and evaluate it at \( P \). The gradient at \( P \) is \( \left.\left\lvert\, \frac{2 \sqrt{3}}{\sqrt{11}}\right.,-\frac{2}{\sqrt{11}}\right) \). (Type exact answers, using radicals as needed.) b. Find the angles \( \theta \) (with respect to the positive \( x \)-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associated with the direction of maximum increase? \( \frac{11 \pi}{6} \) (Type any angles in radians between 0 and \( 2 \pi \). Type an exact answer, using \( \pi \) as needed. Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? \( \square \) (Type any angles in radians between 0 and \( 2 \pi \). Type an exact answer, using \( \pi \) as needed. Use a comma to separate answers as needed.)

To find the angles associated with maximum increase, maximum decrease, and zero change, we start with the gradient we calculated. For maximum increase, the direction of the gradient is given by the angle \( \theta \) which corresponds to the gradient vector. In your case, the gradient at point \( P \) is already evaluated, and the angle \( \theta \) can be found using: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] Thus, plug in the components of the gradient, where the x-component is \( \frac{2 \sqrt{3}}{\sqrt{11}} \) and the y-component is \( -\frac{2}{\sqrt{11}} \). The angle associated with the maximum increase: \[ \theta = \tan^{-1}\left(-\frac{\frac{2}{\sqrt{11}}}{\frac{2\sqrt{3}}{\sqrt{11}}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \] This corresponds to: \[ \theta = \frac{11\pi}{6} \text{ (which is in the fourth quadrant)} \] For maximum decrease, the angle is directly opposite to the gradient direction: \[ \theta + \pi = \frac{11\pi}{6} + \pi = \frac{11\pi}{6} + \frac{6\pi}{6} = \frac{17\pi}{6} \] To fit this within \( [0, 2\pi) \): \[ \frac{17\pi}{6} - 2\pi = \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6} \] Thus, the angles associated with maximum increase and maximum decrease are: - Maximum Increase: \( \frac{11\pi}{6} \) - Maximum Decrease: \( \frac{5\pi}{6} \) - Angles with zero change are any angles perpendicular to these direction angles, which can be calculated as \( \frac{k\pi}{2} \) for integer \( k \). Therefore, your answers are: Maximum increase: \( \frac{11\pi}{6} \) Maximum decrease: \( \frac{5\pi}{6} \)