Question

Determine two positive numbers whose sum is 15 and the sum of whose squares is minimum.

Answer 1

Hint: First from the condition form equation in two variables, and then make a function in single variable and to make it minimum calculate x for\[f'(x) = 0\]. And hence, from there we can calculate the value of one variable, and if the value of one is known the value of another variable can be calculated.

Complete step by step answer:

The given two positive number whose sum is 15 and the sum of whose squares is minimum
Let x and y be two numbers
So, \[x + y = 15\]
Also given the sum of the squares is minimum so \[f(x)\] can be formed as \[f(x) = {x^2} + {y^2}\]
Now, replacing \[y = 15 - x\] in \[f(x)\]
So, \[f(x) = {x^2} + {\left( {15 - x} \right)^2}\]
Now, calculate the value of x by differentiating it equating it to zero.
\[ \Rightarrow \]\[f'(x) = 2x - 2\left( {15 - x} \right) = 0\]
Hence, on calculating the value of x so,
\[ \Rightarrow \]\[4x = 30\]
Hence, on dividing we get,
\[ \Rightarrow \]\[x = \dfrac{{15}}{2}\]
Now, calculating the value y using equation \[x + y = 15\],
Hence, \[y = 15 - \dfrac{{15}}{2}\]
On simplification, we get,
\[ \Rightarrow \]\[y = \dfrac{{15}}{2}\]
Hence, \[\left( {x,y} \right) = \left( {\dfrac{{15}}{2},\dfrac{{15}}{2}} \right)\] is our required answer.

Note: The minimum value of a function is the place where the graph has a vertex at its lowest point. In the real world, you can use the minimum value of a quadratic function to determine the minimum cost or area. The maximum value of a function is the place where a function reaches its highest point, or vertex, on a graph. Practically, finding the maximum value of a function can be used to determine maximum profit or maximum area.