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Let's solve the given problem step-by-step.

### Problem Recap:
We are asked to find the **exact global maximum and minimum** values for the function:
\[
f(t) = \frac{5t}{3 + t^2}
\]
over all real numbers \( t \). 

---

### Step 1: Find the derivative \( f'(t) \)
We use the **quotient rule** for differentiation:
\[
\frac{d}{dt} \left( \frac{u(t)}{v(t)} \right) = \frac{u'(t) v(t) - u(t) v'(t)}{[v(t)]^2}
\]
where \( u(t) = 5t \) and \( v(t) = 3 + t^2 \).

1. \( u'(t) = 5 \)
2. \( v'(t) = 2t \)

Now, apply the quotient rule:
\[
f'(t) = \frac{5(3 + t^2) - 5t(2t)}{(3 + t^2)^2}
\]
Simplify the numerator:
\[
5(3 + t^2) - 10t^2 = 15 + 5t^2 - 10t^2 = 15 - 5t^2
\]
So:
\[
f'(t) = \frac{15 - 5t^2}{(3 + t^2)^2}
\]

---

### Step 2: Find critical points by setting \( f'(t) = 0 \)
We need to solve:
\[
15 - 5t^2 = 0
\]
\[
5t^2 = 15 \quad \Rightarrow \quad t^2 = 3 \quad \Rightarrow \quad t = \pm \sqrt{3}
\]
So, the critical points are \( t = \sqrt{3} \) and \( t = -\sqrt{3} \).

---

### Step 3: Evaluate \( f(t) \) at the critical points and at the limits \( t \to \pm \infty \)
1. **At \( t = \sqrt{3} \)**:
   \[
   f(\sqrt{3}) = \frac{5\sqrt{3}}{3 + (\sqrt{3})^2} = \frac{5\sqrt{3}}{3 + 3} = \frac{5\sqrt{3}}{6}
   \]

2. **At \( t = -\sqrt{3} \)**:
   \[
   f(-\sqrt{3}) = \frac{5(-\sqrt{3})}{3 + (\sqrt{3})^2} = \frac{-5\sqrt{3}}{6}
   \]

3. **As \( t \to \pm \infty \)**:
   \[
   \lim_{t \to \pm \infty} f(t) = \lim_{t \to \pm \infty} \frac{5t}{3 + t^2} = 0
   \]
   So, the function approaches 0 as \( t \) grows very large or very small.

---

### Step 4: Determine the global maximum and minimum
- \( f(\sqrt{3}) = \frac{5\sqrt{3}}{6} \) is the **global maximum**.
- \( f(-\sqrt{3}) = -\frac{5\sqrt{3}}{6} \) is the **global minimum**.

---

### Final Answer
- **Global maximum** at \( t = \sqrt{3} \): \( \frac{5\sqrt{3}}{6} \).
- **Global minimum** at \( t = -\sqrt{3} \): \( -\frac{5\sqrt{3}}{6} \).

---

Do you have any further questions or need more details?  

Here are five related questions to explore:
1. How does the quotient rule for derivatives work?
2. What is the significance of critical points in calculus?
3. Why do we evaluate limits at \( t \to \infty \) for global extrema?
4. How do you know if a critical point is a maximum or minimum?
5. Can the global extrema change if the domain is restricted?

**Tip:** To confirm extrema, you could also use the second derivative test.

Find the exact global maximum and minimum values of the function f(t) = 5t / (3 + t^2) if its domain is all real numbers.

Explore how to find the global maximum and minimum of the function f(t) = 5t / (3 + t^2) over all real numbers. This problem involves key calculus concepts like derivatives, the quotient rule, and limits at infinity. Suitable for advanced high school or early college calculus students.

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