Applications of Derivatives
Minimum value of trigonometric functions
Grade 12
Question:
<p>If \(f(x) = \sec^2 x + 4\,\text{cosec}^2\,x\), then the minimum value of \(f(x)\) is:</p>
<p>7</p>
<p>8</p>
<p>9</p>
<p>10</p>
Step-by-Step Solution
Key Concept: Use the substitution t = tan²x to convert this into a single-variable optimization problem, then apply AM-GM inequality or calculus to find the minimum of the resulting expression.
<p><strong>Step 1:</strong> Express f(x) in terms of a single variable using sec²x = 1 + tan²x and cosec²x = 1 + cot²x.</p><p>Let t = tan²x, where t > 0. Then cot²x = 1/t, so:</p><p>f(x) = (1 + t) + 4(1 + 1/t) = 1 + t + 4 + 4/t = 5 + t + 4/t</p><p><strong>Step 2:</strong> Find the minimum of g(t) = t + 4/t for t > 0 using AM-GM inequality:</p><p>t + 4/t ≥ 2√(t · 4/t) = 2√4 = 4</p><p>Equality holds when t = 4/t, giving t = 2.</p><p><strong>Step 3:</strong> Therefore, the minimum value of f(x) = 5 + 4 = <strong>9</strong>.</p><p>This occurs when tan²x = 2, or equivalently sin²x = 2/3 and cos²x = 1/3.</p><p>∴ Answer: <strong>9</strong></p>
Correct Answer: C