Question

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

Answer 1

Let two numbers be x and y then

x + y = 5  ...(i)

Let S = x³ + y³  ...(ii)

= x³ + (5 – x)³  ...[From (i)]

${\displaystyle \frac{dS}{dx}}$ = 3x² + 3(5 – x)² (– 1)

${\displaystyle \frac{dS}{dx}}$ = 3x² – 3(25 + x² – 10x)

= 3x² – 75 – 3x² + 30x

= 30x – 75

For maximum or minimum

${\displaystyle \frac{dS}{dx}}$ = 0

${\displaystyle \Rightarrow }$ 30x – 75 = 0

${\displaystyle \Rightarrow }$ x = ${\displaystyle \frac{75}{35}=\frac{5}{2}}$

When x = ${\displaystyle \frac{5}{2}}$, y = ${\displaystyle 5-\frac{5}{2}=\frac{5}{2}}$  ...[From (i)]

${\displaystyle \frac{d^{2}S}{{dx}^2}}$ = 30 which is +ve.

So the sum is least when x = ${\displaystyle \frac{5}{2}}$ and y = ${\displaystyle \frac{5}{2}}$

S = x² + y²

= ${\displaystyle \frac{25}{4}+\frac{25}{4}}$

= ${\displaystyle \frac{50}{4}}$

= ${\displaystyle \frac{25}{2}}$