The sum of two positive numbers is given. If the sum of their cubes is minimum, then

Step-by-step solution

Let x and y be two positive numbers. According to question, x + y = a (constant) Let z=x³+y³z=x³+(a-x)³ (i) On differentiating Eq. (i) w.r.t. ' x ', we get aligned dz dx &=3 x ²+3( a - x )²(-1) &=3 [ x ²-( a - x )² ] &=3( x + a - x )( x - a + x ) &=3 a (2 x - a ) aligned For maximum or minimum, dz dx =0 3 a (2 x - a )=0 2 x - a =0 x = a 2 On differentiating Eq. (ii) w.r.t ' x ', we get aligned d ² z dx =& 6 a & d ² z dx ² >0 aligned z has a minimum value at x= a 2 . Therefore, the numbers are a 2 and a- a 2 = a 2 .