WebWork 15.3 #7

The problem is something like the following: The maximum value of $f(x,y)$ subject to the constraint $g(x,y)=250$ is 6000. The method of Lagrange multipliers gives $\lambda=35$. Find an approximate value for the maximum of $f(x,y)$ subject to the constraint $g(x,y)=252$.

Hint: The main idea here is that the functions satisfy the relation $\nabla f=35\nabla g$ at the maximum point where $g(x,y)=250$ and, very importantly, this implies that the gradients are parallel. Hence, if one increases $g$ by 2, one will increase $f$ by some very easy to calculate value. How fast is $f$ increasing compared to $g$??

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