Chapter 13.6, Problem 91ES

(a) Use a CAS to graph $f\left(x,y\right)=\left(x^2+3y^2\right)e^{-\left(x^2+y^2\right)}.$

(b) At how many points do you think it is true that $D_{\text{u}}f\left(x,y\right)=0$ for all unit vectors u?

(c) Use a CAS to find $\nabla f.$

(d) Use a CAS to solve the equation $\nabla f\left(x,y\right)=0\text{for}x\text{and}y.$

(e) Use the result in part (d) together with Theorem 13.6.5 to check your conjecture in part (b).