Answer
(a) $-\nabla f(\bf {x})$
(b) $\lt -12,92 \gt$
Work Step by Step
(a) Formula to calculate the directional derivative: $D_uf=\nabla f(x,y) \cdot
u$
or, $D_uf=|\nabla f(x,y)||u| \cos \theta$
Minimum value of $\cos \theta=-1$ when $\theta =\pi or 180^\circ$
Thus, $D_uf=-|\nabla f(x)|$ or, $-\nabla f(\bf {x})$
(b) Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$
$\nabla f(x,y)=\lt 4x^3y-2xy^3,x^4-3x^2y^2 \gt$
$\nabla f(2,-3)=\lt 12,-92 \gt$
$|\nabla f(2,-3)|=\sqrt{(12)^2+(-92)^2}=\sqrt {\dfrac{9}{25}}=\dfrac{3}{5}$
From part (a) $D_uf=-|\nabla f(x)|=-\lt 12,-92 \gt =\lt -12,92 \gt$
Hence, the required answer is:
$\lt -12,92 \gt$
