Question

The sum of squares of two parts of a number 100 is minimum, then two parts are

• A. 50,50
• B. 25,75
• C. 40,60
• D. 30,70

Answer 1

The correct option is A 50,50

Sum of square of two parts of a number $100$,

Let, the required number are $x$ and $100-x$

Now, according to the question,

$f\left(x\right)=x^2+{(100-x)}^2$ ………(1)

Differentiate with respect to x,

$f^{{}^{\prime }}\left(x\right)=2x+2(100-x)(-1)$ ……..(2)

For maxima and minima,

$f^{{}^{\prime }}\left(x\right)=0$

$2x+2(100-x)(-1)=0$

$4x=200$

$x=50$

Differentiate equation 2^nd with respect to x,

$f^{{}^{\prime \prime }}\left(x\right)=2-2(0-1)$

$f^{{}^{\prime \prime }}\left(x\right)=4$

At $x=50$,

$f^{{}^{\prime \prime }}\left(x\right)>0$

Hence the function $f\left(x\right)=x^2+{(100-x)}^2$ is minimum.

So, required number are $50$ and $50$.

Hence, this is the answer.