A problem in multivariable calculus
The following problem was the first introductory question in a recent exam on multivariable calculus at a top UK university. This question was designed for students who had found the course hard. Indeed you could get a full mark within few seconds using only basic maths from school.
Q1. Find the minimum value of the function \(f(x,y)=x^2+2y^2-4x-4y\) over all real \(x,y\).
Derivatives may not be needed at all
Rather surprisingly, only about 2% of the 200+ class remembered the lecturer’s advice to use appropriate tools. The remaining 98% preferred the hard way finding extreme points through partial derivatives. Similarly to 1-variable functions, as we discussed in the post on global extrema, we may often find extreme values of a function without using any derivatives.
Moreover, about 50% of all students claimed that their critical point is a local minimum without justifications, hence lost almost a half of the full mark. The other 48% spent a lot of time justifying that the only critical point (where both 1st order partial derivatives vanish) is indeed a local minimum by using the Hessian of four 2nd order partial derivatives. About 10% of the class actually wrote more than 2 pages on this first question, though only a couple of lines was enough.
Here is the full solution:
\(f(x,y)=x^2+2y^2-4x-4y=(x-2)^2+2(y-1)^2-6\geq -6\) for all real \(x,y\).
Hence the minimum value is \(f(2,1)=-6\).
This simple technique of completing a square illustrates the powerful principle that mastering the basics is more important than trying an advanced method without proper understanding.
Completing a square vs differentiation
Here is the simple mnemonic rule: What can you do with a quadratic polynomial? – Complete a square, of course! Most students usually try to directly write the roots of a quadratic polynomial by using the so-called quadratic formula. Actually, this formula is proved by completing a square, which is a good exercise especially if you have never done it.
From the computational point of view, the quadratic formula is equivalent to completing a square, so we essentially make the same computations in both cases. However, after completing a square, a quadratic polynomial becomes structured (similar to the simplest form \(x^2\)).
Moreover, the same idea of completing a binom \((x+a)^n\) works for any higher degree polynomial, but the quadratic formula doesn’t. Actually many real problems in mathematics are about putting various objects (say, functions or matrices or groups) into a normal or structured form.
So completing a square is simple and efficient like a hammer, while differentiation is powerful and delicate like a microscope. A popular student question: “Can I still use a microscope, because I like it?” Our answer: “Yes, you can, though it may look a bit unprofessional.”
- Riddle 19: find the maximum value of \(f(x)=6x^3-x^6\) over all real \(x\).
- How to submit: to write your full answer, submit a comment.
- Hint: find a point \(a\) such that \(f(x)\leq f(a)\) for any real \(x\).
- Warning: using derivatives is possible, but is hard to justify.
- Prize: free 1-year access to one of our interactive web tutorials.
- Restriction: only the first correct public answer will be rewarded.
- Update: Carlo has solved the problem, see attempts 1 and 2.
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