Therefore, the absolute value $ |2\sin x + 3\cos x + 4| $ achieves its minimum at the minimum of the continuous function on an interval where it is always positive.

["Title:\nFinding the Minimum of $ |2\sin x + 3\cos x + 4| $: Where the Expression Achieves Its Minimum on a Positive Interval", "Meta Description:\nExplore the minimum value of $ |2\sin x + 3\cos x + 4| $ by analyzing the continuous function $ f(x) = 2\sin x + 3\cos x + 4 $. Learn why this expression reaches its minimum at points where positivity holds, and how understanding amplitude and phase shifts aids in optimization.", "---", "### Understanding $ |2\sin x + 3\cos x + 4| $", "The function $ f(x) = 2\sin x + 3\cos x + 4 $ is a linear combination of sine and cosine functions, both of which oscillate continuously between fixed bounds. The absolute value transforms this periodic expression into a non-negative function, making its minimum meaningful and well-defined.", "Step 1: Rewrite the Trigonometric Expression\nThe core expression $ 2\sin x + 3\cos x $ can be rewritten in the amplitude-phase form:", "$$\n2\sin x + 3\cos x = R \sin(x + \phi)\n$$", "where $ R = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $. The phase shift $ \phi $ satisfies:", "$$\n\cos \phi = \frac{2}{\sqrt{13}}, \quad \sin \phi = \frac{3}{\sqrt{13}}\n$$", "Thus,", "$$\n2\sin x + 3\cos x = \sqrt{13} \sin(x + \phi)\n$$", "So the full function becomes:", "$$\nf(x) = \sqrt{13} \sin(x + \phi) + 4\n$$", "Step 2: Determine the Range of $ f(x) $\nSince $ \sin(x + \phi) $ oscillates between $-1$ and $1$, the expression $ \sqrt{13} \sin(x + \phi) + 4 $ ranges from:", "$$\n4 - \sqrt{13} \quad \ ext{to} \quad 4 + \sqrt{13}\n$$", "Note that $ \sqrt{13} \approx 3.605 $, so:", "$$\nf(x) \in [4 - 3.605,\ 4 + 3.605] \approx [0.395,\ 7.605]\n$$", "Step 3: Analyze the Absolute Value\nBecause $ f(x) = \sqrt{13}\sin(x + \phi) + 4 $ is always greater than or equal to approximately $ 0.395 $, it is always positive on $ \mathbb{R} $. Therefore, $ |f(x)| = f(x) $ everywhere.", "Thus, minimizing $ |2\sin x + 3\cos x + 4| $ is equivalent to minimizing $ f(x) = \sqrt{13} \sin(x + \phi) + 4 $.", "Step 4: Locate the Minimum of $ f(x) $\nThe minimum value $ 4 - \sqrt{13} $ occurs when:", "$$\n\sin(x + \phi) = -1 \quad \Rightarrow \quad x + \phi = \frac{3\pi}{2} + 2\pi k,\quad k \in \mathbb{Z}\n$$", "At these points, the function attains its minimum, and since $ f(x) > 0 $ everywhere, the absolute value achieves its minimum precisely at this minimum.", "Why It’s Always Positive?\nBecause the lowest value $ f(x) \approx 0.395 > 0 $, the absolute value never dips below zero, reinforcing that $ |f(x)| $ achieves its global minimum exactly at the minimum of the continuous function $ f(x) $.", "Conclusion\nThe absolute value $ |2\sin x + 3\cos x + 4| $ reaches its minimum at the critical points of $ f(x) = \sqrt{13} \sin(x + \phi) + 4 $, where the inner expression achieves its negative peak. Since this minimum is above zero, $ |f(x)| $ attains its smallest value precisely there — fully supported by the positive nature of $ f(x) $ over all real numbers.", "---", "### Practical Insight: Using Phase Shifts and Amplitude\nUnderstanding trigonometric identities and amplitude-phase forms enables efficient analysis of such functions. For optimization involving $ |A\sin x + B\cos x + C| $, rewriting the sinusoidal part in amplitude-phase form reveals the extremal values directly, especially when the resultant function remains positive.", "---", "### Related Keywords:\n- Minimum of $ |2\sin x + 3\cos x + 4| $\n- Amplitude-phase form $ 2\sin x + 3\cos x = \sqrt{13}\sin(x+\phi) $\n- Absolute value optimization\n- Continuous function minimum on positive interval\n- Trigonometric optimization techniques", "---", "Keywords: $ |2\sin x + 3\cos x + 4| $, minimum of sinusoidal function, amplitude-phase form, $ \sqrt{13} \sin(x + \phi) $, continuous function optimization, phase shift $ \phi $, $ \sin(x + \phi) = -1 $, positive trigonometric expressions, extremal values, Fourier-based analysis", "---", "Summary:\nBy transforming $ 2\sin x + 3\cos x $ into $ \sqrt{13} \sin(x + \phi) $, we find the function oscillates between about $ 0.395 $ and $ 7.605 $, always positive. Hence, $ |2\sin x + 3\cos x + 4| $ achieves its minimum exactly at the minimum of $ \sqrt{13} \sin(x + \phi) + 4 $, where $ f(x) > 0 $, making optimization both tractable and exact."]