For this problem, consider the function, f (x) = ln (x^2 + 6x + 10)

a) Find all the critical numbers of f(x).

(b) Determine the intervals over which the function is increasing/decreasing.

(c) Classify each critical point as a local maximum/local minimum.

(d) Determine the intervals over which the function is concave up/concave down.

(e) Find all the inflection points of f(x).

Community Answer

【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/5a) To find critical numbers, we will find a derivative and set it equal to zero:\( \begin{align*} \mathrm{{f}'{\left({x}\right)}} &= \mathrm{{0}} \end{align*} \)ExplanationCritical points are the points from where function start to change it's direction Explanation:Please refer to solution in this step.Step2/5b) A function is increasing when:\( \begin{align*} \mathrm{{f}'{\left({x}\right)}} &> \mathrm{{0}} \end{align*} \)A function is decreasing when:\( \begin{align*} \mathrm{{f}'{\left({x}\right)}} &Explanation:Please refer to solution in this step.Step3/5c) To classify, first we will find the second derivative.A critical point is local maximum when:\( \begin{ali ... See the full answer