Find the inflection point(s) of $f ( x ) = \frac { x ^ { 3 } } { 3 } - \frac { 5 } { 2 } x ^ { 2 } + 6 x - 36$
A) $\left( \frac { 5 } { 2 } , \mathrm { f } \left( \frac { 5 } { 2 } \right) \right.$
B) (2, f(2)) and (3, f(3))
C) x = -2, -3
D) (10, f (10))
E) $\left( \frac { 2 } { 5 } , \mathrm { f } \left( \frac { 2 } { 5 } \right) \right.$

Question

Quizplus · Accepted Answer

To find the inflection points of the function $f(x) = \frac{x^3}{3} - \frac{5}{2}x^2 + 6x - 36$, we need to find where the second derivative of $f(x)$ changes sign. First, find the first derivative $f'(x)$:$$f'(x) = x^2 - 5x + 6$$Then, find the second derivative $f''(x)$:$$f''(x) = 2x - 5$$Set $f''(x) = 0$ to find potential inflection points:$$2x - 5 = 0 \Rightarrow x = \frac{5}{2}$$This is the only critical point for the second derivative, so we check the sign of $f''(x)$ around $x = \frac{5}{2}$ to confirm it is an inflection point. Since the sign of $f''(x)$ changes at $x = \frac{5}{2}$, this is indeed an inflection point. Therefore, the correct answer is $\left( \frac{5}{2}, f\left(\frac{5}{2}\right) \right)$.

Find the Inflection Point(s) Of f(x)=x33−52x2+6x−36f ( x ) = \frac { x ^ { 3 } } { 3 } - \frac { 5 } { 2 } x ^ { 2 } + 6 x - 36f(x)=3x3​−25​x2+6x−36

Find the Inflection Point(s) Of $f ( x ) = \frac { x ^ { 3 } } { 3 } - \frac { 5 } { 2 } x ^ { 2 } + 6 x - 36$