<p>Minimum value of <span>\(\log_3\left(\frac{5\sin x - 12\cos x + 26}{13}\right)\)</span> is</p>
Step-by-Step Solution
Key Concept: To minimize a logarithm, we need to minimize its argument. The expression 5sin x - 12cos x can be rewritten in the form R·sin(x + φ), where R = √(5² + 12²) = 13, allowing us to find the minimum value of the entire fraction.
<p><strong>Step 1: Express 5sin x - 12cos x in standard form</strong></p><p>We write 5sin x - 12cos x = R·sin(x + φ) where:</p><p>R = √(5² + (-12)²) = √(25 + 144) = √169 = 13</p><p></p><p><strong>Step 2: Determine the range of the linear combination</strong></p><p>Since 5sin x - 12cos x = 13·sin(x + φ) for some phase angle φ, the range is [-13, 13].</p><p>Therefore: min(5sin x - 12cos x) = -13</p><p></p><p><strong>Step 3: Find the minimum of the numerator</strong></p><p>The numerator is: 5sin x - 12cos x + 26</p><p>Minimum value = -13 + 26 = 13</p><p></p><p><strong>Step 4: Evaluate the minimum of the fraction</strong></p><p>$$\frac{5\sin x - 12\cos x + 26}{13} \geq \frac{13}{13} = 1$$</p><p></p><p><strong>Step 5: Calculate the minimum of the logarithm</strong></p><p>$$\log_3\left(\frac{5\sin x - 12\cos x + 26}{13}\right) \geq \log_3(1) = 0$$</p><p></p><p>The minimum value is attained when 5sin x - 12cos x = -13, which occurs at specific values of x.</p><p></p><p><strong>∴ Answer: Q</strong></p>
Correct Answer: Q