19. let f be the function with derivative defined by f(x)=x³ - 4x. at which of the following values of x…

19. let f be the function with derivative defined by f(x)=x³ - 4x. at which of the following values of x does the graph of f have a point of inflection? (a) 0 (b) 2/3 (c) 2/√3 (d) 4/3 (e) 2

19. let f be the function with derivative defined by f(x)=x³ - 4x. at which of the following values of x does the graph of f have a point of inflection? (a) 0 (b) 2/3 (c) 2/√3 (d) 4/3 (e) 2

Answer

Explanation:

Step1: Recall the condition for inflection point

A function $y = f(x)$ has an inflection - point where $f''(x)=0$ and $f''(x)$ changes sign. First, find the second - derivative of $f(x)$ given $f'(x)=x^{3}-4x$.

Step2: Differentiate $f'(x)$

Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, if $f'(x)=x^{3}-4x$, then $f''(x)=\frac{d}{dx}(x^{3}-4x)=3x^{2}-4$.

Step3: Set $f''(x) = 0$

Set $3x^{2}-4 = 0$. Then $3x^{2}=4$, and $x^{2}=\frac{4}{3}$. Solving for $x$, we get $x=\pm\frac{2}{\sqrt{3}}$.

Answer:

C. $\frac{2}{\sqrt{3}}$