# A patient's pulse measures 110 bpm, 70 bpm, then 60 bpm. To determine an accurate measurement of...

## Question:

A patient's pulse measures 110 bpm, 70 bpm, then 60 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression {eq}(x - 110)^2 + (x - 70)^2 + (x - 60)^2 {/eq}. What value (in bpm) minimizes it?

## Minima and maxima:

To get the extrema of an expression, we equate its first derivative to zero i.e.,

{eq}\frac{df}{dx} = 0 {/eq}

Second derivative test gives whether the extrema is a minima or a maxima.

{eq}\frac{d^2f}{dx^2} < 0 : Maxima \frac{d^2f}{dx^2} > 0 : Minima {/eq}

Given expression {eq}f(x) = (x-110)^2 + (x-70)^2 + (x-60)^2 {/eq}

To find extrema :

{eq}2(x-110) +2(x-70) + 2(x-60) = 0 \\ 3x-240 = 0\\ x=80 {/eq}

Second derivative test :

{eq}f''(x)> 0 {/eq}

Therefore x=80bpm minimizes the expression