Question

The maximum value of $xy$ when $x+2y=8$ is

• A. 20
• B. 16
• C. 24
• D. 8

Hint:

Logic Tip: The AM-GM inequality provides a blazing fast shortcut! For positive numbers $a$ and $b$, $\frac{a+b}{2} \ge \sqrt{ab}$. Let $a=x$ and $b=2y$. Then $\frac{x+2y}{2} \ge \sqrt{2xy}$. Since $x+2y=8$, we have $4 \ge \sqrt{2xy}$. Squaring gives $16 \ge 2xy$, meaning $xy \le 8$. The maximum is clearly 8.

Answer 1

The Correct Option is D

Concept:
This is an optimization problem that can be solved using either the Arithmetic Mean-Geometric Mean (AM-GM) inequality or the application of derivatives. Let's use the derivative method to find the maximum value.
Step 1: Express the function to be maximized in terms of a single variable.
We need to maximize the product $P = xy$. We are given the constraint $x + 2y = 8$. Express $x$ in terms of $y$: $$x = 8 - 2y$$ Substitute this into the product equation: $$P(y) = (8 - 2y)y = 8y - 2y^2$$
Step 2: Find the critical points by setting the first derivative to zero.
Differentiate $P(y)$ with respect to $y$: $$P'(y) = \frac{d}{dy}(8y - 2y^2) = 8 - 4y$$ Set the derivative to zero to find the critical point: $$8 - 4y = 0 \implies 4y = 8 \implies y = 2$$
Step 3: Verify it's a maximum and calculate the final value.
Take the second derivative to verify concavity: $$P''(y) = -4$$ Since $P''(y)<0$, the function has a maximum at $y=2$. Now, find the corresponding value of $x$: $$x = 8 - 2(2) = 8 - 4 = 4$$ Finally, calculate the maximum value of $xy$: $$P_{max} = x \cdot y = 4 \cdot 2 = 8$$