discuss maxima and minima of ( x^3)(y^2)(1x^3)(y^2)(1-x-y)
Answers
When you were learning about derivatives about functions of one variable, you learned some
techniques for finding the maximum and minimum values of functions of one variable. We’ll
now extend those techniques to functions of more than one variable. We’ll concentrate on
functions of two variables, though many of the techniques work more generally.
Local Maxima and Minima
One of the first things you did when you were developing the techniques used to find the
maximum and minimum values of f(x) was you asked yourself
Suppose that the largest (or smallest) value of f(x) is f(a). What does that tell us
about a?
After a little thought you answered
If the largest (or smallest) value of f(x) is f(a) and f is differentiable at a, then f
′
(a) = 0.
Let’s recall what that’s true. Suppose that the largest value of f(x) is f(a). Then for all
h > 0,
f(a + h) ≥ f(a) =⇒ f(a + h) − f(a) ≥ 0 =⇒
f(a + h) − f(a)
h
≥ 0 if h > 0
Taking the limit h → 0 tells us that f
′
(a) ≥ 0. Similarly, for all h < 0,
f(a + h) ≥ f(a) =⇒ f(a + h) − f(a) ≥ 0 =⇒
f(a + h) − f(a)
h
≤ 0 if h < 0
Taking the limit h → 0 now tells us that f
′
(a) ≤ 0. So we have both f
′
(a) ≥ 0 and f
′
(a) ≤ 0
which forces f
′
(a) = 0. You also observed at the time that for this argument to work, you
only need f(x) ≤ f(a) for all x’s close to a, not necessarily for all x’s in the whole world. (In
the above inequalities, we only used f(a + h) with h small.) So you said
If f(a) is a local maximum or minimum for f(x) and f is differentiable at a, then
f
′
(a) = 0.
Exactly the same discussion applies to functions of more than one variable. Here are the
corresponding definitions and statements.
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