Question

$\text{If}S=(7-x{)}^4(x-3{)}^3,x\in (3,7),\text{then}$

• A. $\text{Maximum value of}S=\frac{2^{16}\times 3^3}{7^7}$
• B. $\text{Maximum value attained at}x=\frac{37}{7}$
• C. $\text{Maximum value of}S=\frac{2^{18}\times 3^3}{7^7}$
• D. $\text{Maximum value attained at}x=\frac{33}{7}$

Answer 1

The correct option is D $\text{Maximum value attained at}x=\frac{33}{7}$

$S=(7-x{)}^4(x-3{)}^3\Rightarrow \frac{dS}{dx}=-4(7-x{)}^3(x-3{)}^3+3(7-x{)}^4(x-3{)}^2\Rightarrow \frac{dS}{dx}=0\Rightarrow (7-x{)}^3(x-3{)}^2(-4(x-3)+3(7-x\left)\right)=0\Rightarrow -4x+12+21-3x=0\Rightarrow x=\frac{33}{7}$
The maximum value will be
$S=\frac{{16}^4\times {12}^3}{7^7}\Rightarrow S=\frac{2^{22}\times 3^3}{7^7}$