Solution: To find the maximum of $ F(t) = \frac{100t}{t^2 + 4} $, we compute its derivative using the quotient rule:

["How to Find the Maximum of ( F(t) = \frac{100t}{t^2 + 4} ): A Step-by-Step Guide Using the Quotient Rule", "Finding the maximum value of a function is essential in optimization problems across science, engineering, and economics. One such function is ( F(t) = \frac{100t}{t^2 + 4} ), commonly used in modeling growth and efficiency relationships. In this article, we’ll explore how to find the maximum of ( F(t) ) by computing its derivative using the quotient rule—a fundamental technique in calculus.", "---", "### Understanding the Function", "The function\n[\nF(t) = \frac{100t}{t^2 + 4}\n]\nis a rational function, meaning it’s the ratio of two polynomials. To determine its maximum, we first differentiate it and find critical points where the derivative equals zero or is undefined. The quotient rule provides a reliable method for differentiation of such functions.", "---", "### What is the Quotient Rule?", "The quotient rule states that if you have a function in the form\n[\nf(t) = \frac{u(t)}{v(t)},\n]\nthen its derivative is:\n[\nf'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}.\n]", "For ( F(t) = \frac{100t}{t^2 + 4} ), we identify:\n- ( u(t) = 100t )\n- ( v(t) = t^2 + 4 )", "---", "### Step-by-Step Derivative Calculation", "Step 1: Compute derivatives of ( u(t) ) and ( v(t) )\n[\nu'(t) = \frac{d}{dt}(100t) = 100\n]\n[\nv'(t) = \frac{d}{dt}(t^2 + 4) = 2t\n]", "Step 2: Apply the quotient rule formula", "Substitute into\n[\nF'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}\n= \frac{100(t^2 + 4) - (100t)(2t)}{(t^2 + 4)^2}\n]", "Step 3: Simplify the numerator", "Expand and combine like terms:\n[\n100(t^2 + 4) - 200t^2 = 100t^2 + 400 - 200t^2 = -100t^2 + 400\n]", "So the derivative becomes:\n[\nF'(t) = \frac{-100t^2 + 400}{(t^2 + 4)^2}\n]", "Step 4: Set the derivative equal to zero to find critical points", "[\nF'(t) = 0 \Rightarrow \frac{-100t^2 + 400}{(t^2 + 4)^2} = 0\n]\nSince the denominator is always positive, we only solve the numerator:\n[\n-100t^2 + 400 = 0 \Rightarrow 100t^2 = 400 \Rightarrow t^2 = 4 \Rightarrow t = \pm 2\n]", "---", "### Step 5: Analyze critical points to find maximum", "We have critical points at ( t = 2 ) and ( t = -2 ). Since ( F(t) ) is an odd function multiplied by a linear numerator, and context often relates to positive ( t ), we focus on ( t = 2 ).", "Use the second derivative test or sign analysis:\nPlug values around ( t = 2 ) into ( F'(t) ):\n- For ( t < 2 ) (e.g., ( t = 1 )): ( F'(1) > 0 ) → increasing\n- For ( t > 2 ) (e.g., ( t = 3 )): ( F'(3) < 0 ) → decreasing", "Since ( F(t) ) changes from increasing to decreasing at ( t = 2 ), this point is a local maximum.", "Compute ( F(2) ):\n[\nF(2) = \frac{100 \cdot 2}{2^2 + 4} = \frac{200}{4 + 4} = \frac{200}{8} = 25\n]", "---", "### Final Thoughts on Optimization", "Finding the maximum of functions like ( F(t) ) is critical in real-world applications—from maximizing profit to optimizing machine performance. The quotient rule enables precise differentiation of such functions, paving the way for reliable optimization.", "---", "### Key Takeaways\n✅ Use the quotient rule to differentiate rational functions effectively.\n✅ Identify and analyze critical points to locate maxima.\n✅ Confirm behavior via sign analysis or second derivative tests.\n✅ Interpret results in practical contexts for better decision-making.", "---", "Optimization doesn’t have to be complex. With the right tools like the quotient rule, locating maximum values becomes systematic and powerful.", "---", "For more on calculus techniques, visit our in-depth guides on derivatives and optimization methods."]