Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)?

Question

Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)
f(x) = 2 − 7x^4

(x,y): DNE (answer)

concave upward: DNE (answer)

concave downward: need answer

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Answer 1 · 2018-02-24T07:35:19

$\qquad \qquad "the graph of" \ \ f(x) \ \ "is concave down on" \ \ ( -infty, infty);"$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad "i.e., everywhere."$

$"From what I understand, you want to know about the"$
$"downward concavity of the graph of the function."$
$"(Please correct me if I am wrong !!)"$

$"To investigate the concavity of a function, we will look"$
$"at" \ \ f''(x)."$

$"We are given the function:"$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 2 - 7 x^4.$

$"We compute derivatives:"$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 0 - 7 ( 4 x^3)$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ - 28 x^3.$

$\qquad \qquad :. \qquad \qquad \qquad \qquad \quad \quad \ f''(x) \ = \ - 28 ( 3 x^2 ).$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ f''(x) \ = \ - 84 x^2.$

$"Perhaps before anything else, we note that" \ \ f''(x) \ \ "is clearly"$
$"negative everywhere:"$

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ f''(x) \ < \ 0, \qquad \qquad \qquad \qquad "for all" \ \ x.$

$"Thus, the graph of" \ \ f(x) \ \ "is concave down everywhere:"$

$\qquad \qquad "the graph of" \ \ f(x) \ \ "is concave down on" \ \ ( -infty, infty)."$