Question

The sum of squares of two parts of a number 100 is minimum, then two parts are:
$
  \left( a \right)50,50 \\
  \left( b \right)25,75 \\
  \left( c \right)40,60 \\
  \left( d \right)30,70 \\
 $

Answer 1

Hint: Use application of derivative to find maxima and minima .For maximum and minimum point derivative of function $f'\left( x \right) = \frac{{df}}{{dx}} = 0$ and for check maxima and minima use second derivative test, $f''\left( x \right) > 0$ minima point and $f''\left( x \right) < 0$ maxima point.

Complete step-by-step answer:
Let $x$ and $y$ be two parts of 100.
So, we can write as $x + y = 100$
$ \Rightarrow y = 100 - x$
So, $x$ and $100 - x$ are two parts of 100 .
Now, according to question
$f\left( x \right) = {\left( x \right)^2} + {\left( {100 - x} \right)^2}............\left( 1 \right)$
For maxima and minima, $f'\left( x \right) = \frac{{df}}{{dx}} = 0$ .
So, Differentiate (1) equation with respect to x .
\[
  f'\left( x \right) = \frac{d}{{dx}}\left( {{{\left( x \right)}^2} + {{\left( {100 - x} \right)}^2}} \right) \\
   \Rightarrow f'\left( x \right) = 2x + 2\left( {100 - x} \right)\left( { - 1} \right) \\
   \Rightarrow f'\left( x \right) = 4x - 200..........\left( 2 \right) \\
  f'\left( x \right) = 0 \\
   \Rightarrow 4x - 200 = 0 \\
   \Rightarrow 4x = 200 \\
   \Rightarrow x = 50 \\
\]
Now, use the second derivative test for check x=50 is a maxima or minima point .
So, Differentiate (2) equation with respect to x .
$
  f''\left( x \right) = \frac{d}{{dx}}\left( {4x - 200} \right) \\
   \Rightarrow f''\left( x \right) = 4 \\
 $
$f''\left( x \right) > 0$ for all value of x .
Now, $f''\left( x \right) > 0$ for x=50
 So, x=50 is a minimum point.
Hence the function $f\left( x \right) = {\left( x \right)^2} + {\left( {100 - x} \right)^2}$ minimum at x=50 .
So, the required parts are 50 and 50 .
So, the correct option is (a).

Note: Whenever we face such types of problems we use some important points. First we assume the parts of a number and make a function in one variable according to the question then differentiate the function for maxima and minima then use a second derivative test to confirm the point is maxima or minima.