Question

1)Prove that the function do not have maxima or minima:
$f\left(x\right)=e^x$

2)Prove that the function do not have maxima or minima:
$g\left(x\right)=\log x$

3)Prove that the function do not have maxima or minima:
$h\left(x\right)=x^3+x^2+x+1$

Answer 1

1) Given $f\left(x\right)=e^x$

Differentiating w.r.t $x,$

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

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

$e^x=0$

This is not possible for any $xϵR$

$\therefore f\left(x\right)$ does not have a maxima or minima.

2) Given $g\left(x\right)=\log x$

Differentiation w.r.t $x,$

$g^{\prime }\left(x\right)=\frac{1}{x}$

Putting $g^{\prime }\left(x\right)=0$

$\frac{1}{x}=0$

This is not possible for any $xϵR$

$\therefore g\left(x\right)$ does not have a maxima or minima.

3) given $h\left(x\right)=x^3+x^2+x+1$

Differentiation w.r.t $x,$

$h^{\prime }\left(x\right)=3x^2+2x+1$ Putting $h^{\prime }\left(x\right)=0$

$3x^2+2x+1=0$

Check Discriminant,

$D=b^2-4ac$

$\Rightarrow D=2^2-4\times 3\times 1$

$\Rightarrow D=-8<0$

So, $3x^2+2x+1>0$ for all $xϵR$

$\therefore h\left(x\right)$ does not have a maxima or minima.