Find the inflection point(s) of $f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 16$.
A) $\left( \frac { 7 } { 2 } , \mathrm { f } \left( \frac { 7 } { 2 } \right) \right.$
B) (2, f (2)) and (6, f (6))
C) (56, f (56))
D) (3, f (3)) and (4, f (4))
E) $\left( \frac { 2 } { 7 } , \mathrm { f } \left( \frac { 2 } { 7 } \right) \right.$

Question

Quizplus · Accepted Answer

To find the inflection points of the function $f(x) = \frac{2}{3}x^3 - 7x^2 + 24x - 16$, we need to find where the second derivative changes sign. First, find the first derivative $f'(x)$, then the second derivative $f''(x)$, and solve $f''(x) = 0$.First derivative: $f'(x) = 2x^2 - 14x + 24$.Second derivative: $f''(x) = 4x - 14$.Setting $f''(x) = 0$ gives $4x - 14 = 0$, or $x = \frac{14}{4} = \frac{7}{2}$.This is the point where the concavity of the function changes, making it the inflection point. Therefore, the correct answer is the point at $x = \frac{7}{2}$, which corresponds to choice A.

Find the Inflection Point(s) Of f(x)=23x3−7x2+24x−16f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 16f(x)=32​x3−7x2+24x−16

Find the Inflection Point(s) Of $f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 16$