Question

CALC • Correct me if I get anything wrong in the process... haven't finished this question yet

Find the following:
a. intervals of increase/decrease
"when f'(x) > 0, increasing
when f'(x) < 0, decreasing"
b. local min/max values
"when f'(x) = 0; min, f"(x) > 0 and max, f"(x) < 0"
c. intervals of concavity and inflection points...
"when f"(x) either < 0 or > 0, and when f"(x)=0"

(optional) sketch a graph using this information.

Answer 1

for function: \[f(x)=\ln(x^4+1)\]

Answer 2

part a.\[f'(x)=\frac{1}{x^4+1}\times4x^3=\frac{4x^3}{x^4+1}\]so...\[\frac{4x^3}{x^4+1}\lt0\]
part b.\[\frac{4x^3}{x^4+1}=0\]
part c.\[f''(x)=12x^2(x^4+1)+4x^3((x^4+1)^{-2}\times4x^3)=\frac{12x^2}{x^4+1}+\frac{(4x^3)^2}{(x^4+1)^2}\]

Answer 3

sorry, first part of (C) is (x^4+1)^(-1), which is why it becomes denominator.

and (B) x=0, solved

Answer 4

If I'm following correctly, you found that \(x=0\) is the only critical point for \(f\), which is correct.

It doesn't look like you've completed part (a). You need to solve that inequality, and the solution for \(f'(x)<0\) will tell you where \(f\) is decreasing, while the solution for \(f'(x)>0\) will tell you where it's increasing.

The second derivative is
\[f''(x)=\frac{12x^2(x^4+1)-16x^6}{(x^4+1)^2}=\frac{12x^2-4x^6}{(x^4+1)^2}\]also correct. From here you would need to solve \(f''(x)>0\) and \(f''(x)<0\) to determine concavity, which is easy enough to do but can be made easier by looking for the candidates for inflection points first, then testing the sign of \(f''\) between each of those.
\[\frac{12x^2-4x^6}{(x^4+1)^2}=0\implies 12x^2-4x^6=4x^2(\sqrt[4]3-x)(\sqrt[4]3+x)(\sqrt3+x^2)=0\]So some possible inflection points occur at \(x=\pm\sqrt[4]3\approx\pm1.3161\) and \(x=0\).

It should be easy enough to find where \(f''\) is positive/negative, then determine the function's concavity. Knowing where \(f\) is concave and where it's convex will inform you what kind of extremum \(f\) has at its critical point \(x=0\).

Answer 5

@HolsterEmission Thanks! That was very detailed.
Just a few questions...
A) how do I solve the inequality? That's where I got stuck...
B) how do I know if the x=0 is a maximum or minimum... through calculations, not a graph? Does solving (A) help me with that?