Chapter 11.8, Problem 23E

(a)

To determine

To find: The extreme value of the function $f\left(x,y\right)=x$ subject to the constraint $y^2+x^4-x^3=0$ by using Lagrange multipliers.

(b)

To determine

To show: The minimum value of $f\left(0,0\right)=0$ and find the Lagrange condition $\nabla f\left(0,0\right)=\lambda \nabla g\left(0,0\right)$ does not exist.

(c)

To determine

To explain: The failure of the Lagrange multipliers while finding the minimum values.