let f(x)=x³ + 3x² - 189x + 1. a. use the definition of a derivative or the derivative rules to find f(x)= b…

let f(x)=x³ + 3x² - 189x + 1. a. use the definition of a derivative or the derivative rules to find f(x)= b. use the definition of a derivative or the derivative rules to find f(x)= c. on what interval is f increasing? interval of increasing = d. on what interval is f decreasing? interval of decreasing = e. on what interval is f concave downward? interval of downward concavity = f. on what interval is f concave upward?

let f(x)=x³ + 3x² - 189x + 1. a. use the definition of a derivative or the derivative rules to find f(x)= b. use the definition of a derivative or the derivative rules to find f(x)= c. on what interval is f increasing? interval of increasing = d. on what interval is f decreasing? interval of decreasing = e. on what interval is f concave downward? interval of downward concavity = f. on what interval is f concave upward?

Answer

Explanation:

Step1: Find the first - derivative

Using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $f(x)=x^{3}+3x^{2}-189x + 1$, we have $f'(x)=3x^{2}+6x-189$.

Step2: Find the second - derivative

Differentiate $f'(x)$ with the power rule. So $f''(x)=\frac{d}{dx}(3x^{2}+6x - 189)=6x+6$.

Step3: Find where $f(x)$ is increasing

Set $f'(x)>0$. [ \begin{align*} 3x^{2}+6x-189&>0\ x^{2}+2x - 63&>0\ (x + 9)(x - 7)&>0 \end{align*} ] The critical points are $x=-9$ and $x = 7$. The solution of the inequality is $x<-9$ or $x>7$. So the interval of increasing is $(-\infty,-9)\cup(7,\infty)$.

Step4: Find where $f(x)$ is decreasing

Set $f'(x)<0$. From $(x + 9)(x - 7)<0$, the solution is $-9<x<7$. So the interval of decreasing is $(-9,7)$.

Step5: Find where $f(x)$ is concave downward

Set $f''(x)<0$. [ \begin{align*} 6x+6&<0\ x&<-1 \end{align*} ] So the interval of downward concavity is $(-\infty,-1)$.

Step6: Find where $f(x)$ is concave upward

Set $f''(x)>0$. [ \begin{align*} 6x+6&>0\ x&>-1 \end{align*} ] So the interval of upward concavity is $(-1,\infty)$.

Answer:

a. $f'(x)=3x^{2}+6x - 189$ b. $f''(x)=6x + 6$ c. $(-\infty,-9)\cup(7,\infty)$ d. $(-9,7)$ e. $(-\infty,-1)$ f. $(-1,\infty)$