Question

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

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

Answer 1

Maximum or minimum can be seen by using derivatives.

Steps1: First find first derivative of the function

Step2: Put it equal to zero and find $x$ were first derivative is zero

Step3: Now find second derivative

Step4: Put $x$ for which first derivative was zero in equation of second derivative

Step5: If second derivative is greater than zero then function takes minimum value at that $x$ and if second derivative is negative then function will take maximum value at that $x$. If Second derivative is zero them it means that this is the point of inflection.

So here $g^{{}^{\prime }}\left(x\right)=3x^2$

Putting this equal to $0$, we get
$g^{{}^{\prime }}\left(x\right)=3x^2=0$
$\Rightarrow x=0$.

Second derivative of this function is

$f^{{}^{\prime \prime }}\left(x\right)=6x$
At $x=0$
$f^{{}^{\prime \prime }}\left(0\right)=6\ast 0=0$
Which means this function has no maxima or minima but will show an inflection i.e slope will change from negative to positive or from positive to negative at $x=0$