Problem

PART 2:

2. $f(x,y)$ is a differentiable function. On the curve $\tan^{-1}(xy)+y=2+\frac{\pi}{4}$, $f$ obtains local maximum at $(x,y)=\left(\frac{1}{2},2\right)$. Suppose that $f_x\left(\frac{1}{2},2\right)=2$.

(a) Find $\nabla f\left(\frac{1}{2},2\right)$.

(b) Assume that on another curve $\tan^{-1}(xy)+y=1.9+\frac{\pi}{4}$, $f$ obtains local maximum at $(x_0,y_0)$ which is near $\left(\frac{1}{2},2\right)$. Use linear approximation to estimate $f(x_0,y_0)-f\left(\frac{1}{2},2\right)$.