(a) Find the maximum and minimum of the function f(x,y)=x+y subject to the const
ID: 1892918 • Letter: #
Question
(a) Find the maximum and minimum of the function f(x,y)=x+ysubject to the constraint g(x,y)=xy=1
(b) Find the minimum of the function f(x,y,z)=x2+y2+z2 subject to the
constraints 2y+z=6 and x-2y=4
Explanation / Answer
FUNCTION f(x) has a relative maximum value at x = a, if f(a) is greater than any value in its immediate neighborhood. We call it a "relative" maximum because other values of the function may in fact be greater. We say that a function f(x) has a relative minimum value at x = b, if f(b) is less than any value in its immediate neighborhood. Again, other values of the function may in fact be less. With that understanding, then, we will drop the term relative. The value of the function, the value of y, at either a maximum or a minimum is called an extreme value. Now, what characterizes the graph at an extreme value? The tangent to the curve is horizontal. We see this at the points A and B above. The slope of each tangent line -- the derivative when evaluated at a or b -- is 0. f '(x) = 0. Moreover, at points immediately to the left of a maximum -- at a point C -- the slope of the tangent is positive: f '(x) > 0. While at points immediately to the right -- at a point D -- the slope is negative: f '(x) < 0. In other words, at a maximum, f '(x) changes sign from + to - . At a minimum, f '(x) changes sign from - to + . We can see that at the points E and F. We can also observe that at a maximum, at A, the graph is concave downward. (Topic 14 of Precalculus.) While at a minimum, at B, it is concave upward. A value of x at which the function has either a maximum or a minimum is called a critical value. In the figure, the critical values are x = a and x = b. The critical values determine turning points, at which the tangent is parallel to the x-axis. The critical values -- if any -- will be the solutions to the equation f '(x) = 0.Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.