2. let f(x)=x^4 - 32x. which of the following is true? a ) f(x) has a critical point at x = 2 and no…

2. let f(x)=x^4 - 32x. which of the following is true? a ) f(x) has a critical point at x = 2 and no inflection points b ) f(x) has a critical point at x = 3√4 and no inflection points. c ) f(x) has a critical point at x = 3√4{4} and an inflection point at x = 2. d ) f(x) has a critical point at x = 2 and an inflection point at x = 0

2. let f(x)=x^4 - 32x. which of the following is true? a ) f(x) has a critical point at x = 2 and no inflection points b ) f(x) has a critical point at x = 3√4 and no inflection points. c ) f(x) has a critical point at x = 3√4{4} and an inflection point at x = 2. d ) f(x) has a critical point at x = 2 and an inflection point at x = 0

Answer

Explanation:

Step1: Find the first - derivative

First, find the derivative of $f(x)=x^{4}-32x$. Using the power rule $\frac{d}{dx}(x^{n}) = nx^{n - 1}$, we have $f^\prime(x)=4x^{3}-32$. Set $f^\prime(x) = 0$ to find critical points: $$4x^{3}-32 = 0$$ $$4x^{3}=32$$ $$x^{3}=8$$ $$x = 2$$

Step2: Find the second - derivative

Next, find the second - derivative of $f(x)$. Differentiate $f^\prime(x)=4x^{3}-32$ with respect to $x$. Using the power rule again, $f^{\prime\prime}(x)=12x^{2}$. Set $f^{\prime\prime}(x)=0$ to find inflection points: $$12x^{2}=0$$ $$x = 0$$

Answer:

D. $f(x)$ has a critical point at $x = 2$ and an inflection point at $x = 0$