Prove that the following functions do not have maxima or minima:

Solution:

i. We have,

$f(x)=\mathrm{e}^{\mathrm{x}}$

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

Now, if $f^{\prime}(x)=0$, then $e^{x}=0 .$ But, the exponential function can never assume 0 for any value of $x$.

Therefore, there does not exist $c \in \mathbf{R}$ such that $f^{\prime}(c)=0$.

Hence, function f does not have maxima or minima.

ii. We have,

$g(x)=\log x$

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

Since $\log x$ is defined for a positive number $x, g^{\prime}(x)>0$ for any $x$.

Therefore, there does not exist $c \in \mathbf{R}$ such that $g^{\prime}(c)=0$.

Hence, function g does not have maxima or minima.

iii. We have,

$h(x)=x^{3}+x^{2}+x+1$

$\therefore h^{\prime}(x)=3 x^{2}+2 x+1$

Now,

$h(x)=0 \Rightarrow 3 x^{2}+2 x+1=0 \Rightarrow x=\frac{-2 \pm 2 \sqrt{2} i}{6}=\frac{-1 \pm \sqrt{2} i}{3} \notin \mathbf{R}$

Therefore, there does not exist $c \in \mathbf{R}$ such that $h^{\prime}(c)=0$.

Hence, function h does not have maxima or minima.