SUMMARY
The minimum value of the function \( f(x) = \sqrt{p^2 + x^2} + \sqrt{(q - x)^2 + r^2} \) for positive real numbers \( p, q, r \) occurs at \( x = \frac{q}{2} \) when \( p = r \). This conclusion is derived from applying calculus and geometric interpretations to the problem, confirming that the function is minimized when the distances represented by the square roots are balanced. The minimum value can be computed as \( f\left(\frac{q}{2}\right) = \sqrt{p^2 + \left(\frac{q}{2}\right)^2} + \sqrt{\left(\frac{q}{2}\right)^2 + r^2} \).
PREREQUISITES
- Understanding of calculus, particularly optimization techniques.
- Familiarity with geometric interpretations of functions.
- Knowledge of the properties of square roots and distance calculations.
- Basic algebra skills for manipulating equations.
NEXT STEPS
- Study optimization techniques in calculus, focusing on critical points and second derivative tests.
- Explore geometric interpretations of functions to visualize distance minimization.
- Learn about the Cauchy-Schwarz inequality and its applications in optimization problems.
- Investigate similar problems involving distance minimization in multi-variable calculus.
USEFUL FOR
Mathematicians, students studying calculus, and anyone interested in optimization problems involving geometric interpretations and distance calculations.
