What two positive real numbers whose product is 32 have the smallest possible sum? Let S be the sum of the two numbers. What is the objective function in terms of one number, x ? $$ \mathrm{x}+\frac{32}{\mathrm{x}} $$ (Type an expression.) The interval of interest of the function is $$ \square $$. (Simplify your answer. Type your answer in interval notation.) The numbers are $$ \square $$ and $$ \square $$. (Type exact answers, using radicals as needed.)

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UpStudy AI · Accepted Answer

**Objective Function:**
$$
x + \frac{32}{x}
$$

**Interval of Interest:**
$$
(0, \infty)
$$

**The Numbers:**
$$
4\sqrt{2} \quad \text{and} \quad 4\sqrt{2}
$$

---

**Explanation:**

1. **Objective Function:**
   - Let the two positive real numbers be $$ x $$ and $$ y $$ with the constraint $$ xy = 32 $$.
   - The sum $$ S = x + y $$. To express this in terms of one variable, solve for $$ y $$: $$ y = \frac{32}{x} $$.
   - Thus, the objective function in terms of $$ x $$ is:
     $$
     S(x) = x + \frac{32}{x}
     $$

2. **Interval of Interest:**
   - Since both numbers are positive real numbers, $$ x $$ must be greater than 0.
   - Therefore, the interval of interest is:
     $$
     (0, \infty)
     $$

3. **Finding the Numbers with the Smallest Sum:**
   - To minimize $$ S(x) = x + \frac{32}{x} $$, take the derivative and set it to zero:
     $$
     S'(x) = 1 - \frac{32}{x^2} = 0 \implies x^2 = 32 \implies x = \sqrt{32} = 4\sqrt{2}
     $$
   - Since $$ x = y $$ for the minimum sum (by AM ≥ GM inequality), both numbers are:
     $$
     4\sqrt{2} \quad \text{and} \quad 4\sqrt{2}
     $$

These values ensure that the product is 32 and the sum is minimized.
**Objective Function:** $$ x + \frac{32}{x} $$ **Interval of Interest:** $$ (0, \infty) $$ **The Numbers:** $$ 4\sqrt{2} \quad \text{and} \quad 4\sqrt{2} $$

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