Question

# Solve and give the correct answer with using second derivative of the function as follows displaystyle f{{left({x}right)}}={left({x}+{9}right)}^{2}

Modeling data distributions
Solve and give the correct answer with using second derivative of the function as follows $$\displaystyle f{{\left({x}\right)}}={\left({x}+{9}\right)}^{2}$$

2021-02-13
The given function is $$\displaystyle f{{\left({x}\right)}}{i}{s}{\left({x}+{9}\right)}^{2}.$$
Obtain the first and second derivative of the function as follows.
$$\displaystyle f{{\left({x}\right)}}={\left({x}+{9}\right)}^{2}$$
$$\displaystyle{f}'{\left({x}\right)}={2}{\left({x}+{9}\right)}$$
$$\displaystyle{f}'{\left({x}\right)}={2}$$
Find the critical points as follows
$$\displaystyle{f}'{\left({x}\right)}={0}$$
$$\displaystyle{2}{\left({x}+{9}\right)}={0}$$
$$\displaystyle\frac{{{2}{\left({x}+{9}\right)}}}{{2}}=\frac{0}{{2}}$$
$$\displaystyle{x}=-{9}$$
Thus, the function has critical point at $$\displaystyle{x}=-{9}$$
Since the second derivative is a constant function and is greater than zero for all values of x, the function has only local minima.
Thus, the function has local minima or minimum at $$\displaystyle{\left(–{9},{0}\right)}.$$