Question

If $x$ is a positive real number, then the greatest value of $(7-x)(x+5{)}^2$ is

• A. $256$
• B. $128$
• C. $64$
• D. $40$

Answer 1

Answer: (D)

$40$

Given, $(7-x)(x+5{)}^2$

taking the derivative and equating to zero

$⟹2(7-x)(x+5)+(x+5{)}^2(7-x\left)\right(-1)=0$

$⟹(7-x)(x+5)(2-(x+5\left)\right)=0$

$⟹(7-x)(x+5)(-x-3)=0$

$⟹(7-x)=0$ and $(x+5)=0$ and $(-x-3)=0$

$⟹x=7$ and $x=-5$ and $x=-3$

substituting $x=7⟹(7-7)(7+5{)}^2=0$

substituting $x=-5⟹(7+5)(-5+5{)}^2=0$

substituting $x=-3⟹(7+3)(-3+5{)}^2=10\times 4=40$