How to find the maxima and minima of y=x+1/x

A very easy question

Proceed this way

We have[math] \displaystyle = x+ \frac{1}{x}[/math]

[math] \displaystyle y'= \frac{dx}{dx}+ \frac{d \left( \frac{1}{x} \right) }{dx}[/math]

[math]\displaystyle y'= 1 - \frac{1}{x^2}[/math]

Note: At minima and maxima y' = 0

[math]\displaystyle 1 - \frac{1}{x^2} =0[/math]

[math]\displaystyle 1= \frac{1}{x^2}[/math]

[math]\displaystyle x^2 =1[/math]

[math]\displaystyle x=1 \quad \text{or} \quad x=-1[/math]

CHECKING FOR MAXIMA AND MINIMA

[math]\displaystyle y''= \frac{d 1}{dx} -\frac{d \left( \frac{1}{x^2} \right)}{dx}[/math]

[math]\displaystyle y'' = 0 +2 \frac{1}{x^3}[/math]

Putting in values

[math]\displaystyle x=1 \rightarrow \frac{1}{1^3}[/math]

[math]\displaystyle 1 > 0 \rightarrow y'' >0 \qquad \text{HENCE MINIMA}[/math]

THUS MINIMA [math]\displaystyle y = 1+ \frac{1}{1}=2[/math]

[math]\displaystyle x=-1 \rightarrow \frac{1}{ \left( -1 \right) ^3}[/math]

[math]\displaystyle -1<0 \rightarrow y''<0 \qquad \text{HENCE MAXIMA}[/math]

THUS MAXIMA [math]\displaystyle y=1 + \frac{1}{-1} = 0[/math]

Hope it helps