Please show all parts. I want to understand. including the concavity chart and the sign chart and especially how you find the inflection point.

$f (x) = e^{1 ? 2 x^{3}}$ (a) Find all cut points of the second derivative, $f^{??} (x)$ : (b) Give the sign chart of $f^{??} (x)$ (c) Give the corresponding concavity chart of $f (x)$ : (d) Give the interval(s) where the function $f (x)$ is concave upward: (e) Give the interval(s) where the function $f (x)$ is concave downward: (f) Give the inflection point (s) in the format $(x, y)$ :

Question

Please show all parts. I want to understand. including the concavity chart and the sign chart and especially how you find the inflection point.

f (x) = e^{1 ? 2 x^{3}}

(a) Find all cut points of the second derivative,

f^{??} (x)

: (b) Give the sign chart of

f^{??} (x)

(c) Give the corresponding concavity chart of

f (x)

: (d) Give the interval(s) where the function

f (x)

is concave upward: (e) Give the interval(s) where the function

f (x)

is concave downward: (f) Give the inflection point (s) in the format

(x, y)

:

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