Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
Let =(\boldsymbol{x}-\boldsymbol{p})^{2}+(\boldsymbol{x}-\boldsymbol{q})^{2}+(\b
Let \( \boldsymbol{f}(\boldsymbol{x})=(\boldsymbol{x}-\boldsymbol{p})^{2}+(\boldsymbol{x}-\boldsymbol{q})^{2}+(\boldsymbol{x}- \) \( \boldsymbol{r})^{2} . \) Then \( \boldsymbol{f}(\boldsymbol{x}) \) has a minimum at \( \boldsymbol{x}=\boldsymbol{\lambda} \) where \( \lambda \) is equal to
- A. \( \frac{p+q+r}{3} \)
- B. \( \sqrt[3]{p q r} \)
- C. \( \frac{3}{\frac{1}{p}+\frac{1}{q}+\frac{1}{r}} \)
- D. none of these
Step-by-step solution
The function is a quadratic in x with positive coefficient (3), so it attains a minimum at its vertex. Setting derivative f'(x) = 2(x-p)+2(x-q)+2(x-r) = 2(3x-(p+q+r)) = 0 gives x = (p+q+r)/3.
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