Question

How do you find the local extremas for $g(x)={{x}^{2}}+1$ ?

Answer 1

To find the local extrema of the given function first we will differentiate the given function with respect to x to find its first order derivative. Then we will equate the first order derivative with zero to find the point of local extrema.

Complete step-by-step answer:
We have been given a function $g(x)={{x}^{2}}+1$.
We have to find the local extremas for the given function.
We know that local extrema of a function is the point at which a function has maximum or minimum value.
Now, first we will differentiate the given function with respect to x to find the first order derivative of the function. Then we will get

& \Rightarrow g(x)=\dfrac{d}{dx}\left( {{x}^{2}}+1 \right) \\\ & \Rightarrow g(x)=\dfrac{d}{dx}{{x}^{2}}+\dfrac{d}{dx}1 \\\ \end{aligned}$$ Now, we know that $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ Now, applying the differentiation rule to the above obtained equation we will get $$\begin{aligned} & \Rightarrow g'(x)=2x+0 \\\ & \Rightarrow g'(x)=2x \\\ \end{aligned}$$ Now, equating the first order derivative with zero we will get $\begin{aligned} & \Rightarrow g'(x)=0 \\\ & \Rightarrow 2x=0 \\\ \end{aligned}$ Now, simplifying the above obtained equation we will get $\Rightarrow x=0$ It means the function has local extrema i.e. either minima or maxima at $x=0$ . Hence $x=0$ is the local extrema of a given function. **Note:** The point to be noted is that the first order derivative of a function is the slope of the function. When the slope of the function will be zero the function obtains a maximum or minimum value at that point. In order to find whether it is maxima or minima we need to find the second order derivative of the function.