Question:
<p>If <span class="math-tex">\(\alpha, \beta, \gamma>0\)</span> then the minimum value of the function f(x) = <span class="math-tex">\(\sqrt{\alpha^{2}+x^{2}}+\sqrt{(x-\beta)^{2}+\gamma^{2}}\)</span> is :</p>
<p style="display:inline"><span class="math-tex">\(\sqrt{\gamma^{2}+(\alpha+\beta)^{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\sqrt{\alpha^{2}+(\beta+\gamma)^{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\sqrt{\beta^{2}+(\alpha+\gamma)^{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\alpha+\beta+\gamma\)</span></p>
Step-by-Step Solution
Key Concept: Interpret the function geometrically as the sum of distances from a point (x, 0) to two fixed points (0, -α) and (β, γ), which is minimized when the path is a straight line.
<p><span class="math-tex">$\sqrt{\beta^{2}+(\alpha+\gamma)^{2}}$</span></p>
Correct Answer: C