(Solved): (a) Show that a differentiable function \( f \) decreases most rapidly at \( \mathrm{x} \) in the d ...

(a) Show that a differentiable function \( f \) decreases most rapidly at \( \mathrm{x} \) in the direction opposite the gradient vector, that is, in the direction of \( -\nabla f(x) \) Let \( \theta \) be the angle between \( \nabla f(\mathbf{x}) \) and unit vector \( \mathbf{u} \). Then \( D_{\mathbf{u}} f=|\nabla f| \) of \( D_{u} f \) is \( -|\nabla f| \), occurring when the direction of \( u \) is \( 0 \leq \theta<2 \pi \), when \( \theta= \) (b) Use the result of part (a) to find the direction in which the function \( f(x, y)=x^{3} y-x^{2} y^{3} \) decreases fastest at the point \( (5,-3) \)