To find the maximum value of the product [tex]\( x \cdot y \)[/tex] subject to the constraint [tex]\( x + y = 8 \)[/tex], we can follow these steps:
1. Express one variable in terms of the other using the constraint:
The constraint given is [tex]\( x + y = 8 \)[/tex]. We can solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 8 - x
\][/tex]
2. Substitute this expression into the product function:
Substitute [tex]\( y = 8 - x \)[/tex] into the expression [tex]\( xy \)[/tex]:
[tex]\[
f(x) = x(8 - x)
\][/tex]
Simplify the function:
[tex]\[
f(x) = 8x - x^2
\][/tex]
3. Find the critical points by differentiating and setting the derivative to zero:
Differentiate [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
f'(x) = 8 - 2x
\][/tex]
Set the derivative equal to zero to find the critical points:
[tex]\[
8 - 2x = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4
\][/tex]
4. Determine the corresponding [tex]\( y \)[/tex] value:
Substitute [tex]\( x = 4 \)[/tex] back into the constraint equation [tex]\( x + y = 8 \)[/tex]:
[tex]\[
4 + y = 8
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 4
\][/tex]
5. Evaluate the product [tex]\( xy \)[/tex] at the critical point:
Finally, substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 4 \)[/tex] into the expression [tex]\( xy \)[/tex]:
[tex]\[
xy = 4 \cdot 4 = 16
\][/tex]
So, the critical point is [tex]\((4, 4)\)[/tex] and the maximum value of [tex]\( xy \)[/tex] given the constraint [tex]\( x + y = 8 \)[/tex] is [tex]\( 16 \)[/tex].
