Question

Find the maximum and minimum values, if any, of the following function given by,

$g\left(x\right)=x^3+1$

Answer 1

Given, function is $g\left(x\right)=x^3+1$
We observe that the value of f(x) increases when the value of x increase and f(x) can be made as large as we please by giving large value to x, So, f(x) does not have a maximum value. Similarly, f(x) can be made as small as we please by giving smaller values of x. So, f(x) does not have the minimum value.

Alternate method:

Here, $g\left(x\right)=x^3+1,g^{\prime }\left(x\right)=3x^2,g"\left(x\right)=6x$
For maxima or minima put g'(x)=0
$\Rightarrow 3x^2=0\Rightarrow x=0$
At $x=0,g"\left(0\right)=6\times 0=0$
Hence at x=0, g(x) is neither maxima nor minima. It is a point of inflection.