Question

Examine the function $f\left(x\right)=x+\frac{25}{x}$ for maxima and minima

Answer 1

$\Rightarrow$ $f\left(x\right)=x+\frac{25}{x}$

Taking derivative on both sides,

$\Rightarrow$ $f^{\prime }\left(x\right)=1-\frac{25}{x^2}$ ---- ( 1 )

Critical point :

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

$\Rightarrow$ $1-\frac{25}{x^2}=0$

$\Rightarrow$ $x^2-25=0$

$\Rightarrow$ $x^2=25$

$\therefore$ $x=\pm 5$

Now, taking derivative of ( 1 ), we get

$\Rightarrow$ $f^{\prime \prime }\left(x\right)=\frac{50}{x^3}$

$\Rightarrow$ Now, $f^{\prime \prime }(-5)=\frac{50}{(-5{)}^3}=-\frac{2}{5}$

$\Rightarrow$ $f^{\prime \prime }\left(5\right)=\frac{50}{(5{)}^3}=\frac{2}{5}$

$\therefore$ Maximum value of $f\left(x\right):f(-5)=-\frac{2}{5}$

$\therefore$ Minimum value of $f\left(x\right):f\left(5\right)=\frac{2}{5}$