$\text { Identify the open intervals where the function } f ( x ) = 6 x ^ { 2 } - 6 x + 4 \text { is increasing or }$ decreasing.
A) decreasing on $( - \infty , \infty )$
B) increasing on $( - \infty , \infty )$
C) increasing: $\left( - \infty , \frac { 1 } { 2 } \right)$; decreasing: $\left( \frac { 1 } { 2 } , \infty \right)$
D) decreasing on $( - \infty , 0 )$; increasing on $( 0 , \infty )$
E) decreasing: $\left( - \infty , \frac { 1 } { 2 } \right) ;$ increasing: $\left( \frac { 1 } { 2 } , \infty \right)$

Question

$\text { Identify the open intervals where the function } f ( x ) = 6 x ^ { 2 } - 6 x + 4 \text { is increasing or }$ decreasing. 
A) decreasing on $( - \infty , \infty )$
B) increasing on $( - \infty , \infty )$
C) increasing: $\left( - \infty , \frac { 1 } { 2 } \right)$; decreasing: $\left( \frac { 1 } { 2 } , \infty \right)$
D) decreasing on $( - \infty , 0 )$; increasing on $( 0 , \infty )$
E) decreasing: $\left( - \infty , \frac { 1 } { 2 } \right) ;$ increasing: $\left( \frac { 1 } { 2 } , \infty \right)$

Quizplus · Accepted Answer

To determine where the function is increasing or decreasing, we need to find the critical points of the function, which are the points where the derivative is equal to 0 or undefined. \begin{align*} f(x) &= 6x^2 - 6x + 4 \ f'(x) &= 12x - 6 \end{align*} Setting the derivative equal to 0: \begin{align*} 12x - 6 &= 0 \ 12x &= 6 \ x &= \frac{1}{2} \end{align*} This critical point divides the x-axis into two regions: $\left(-\infty,\frac{1}{2} ight)$ and $\left(\frac{1}{2},\infty ight)$. Now we can test the sign of the derivative in each of these regions: \begin{align*} f'(x) > 0 &: ext{Increasing} \ f'(x) < 0 &: ext{Decreasing} \end{align*} For $x < \frac{1}{2}$, $f'(x) = 12x - 6 < 0$, so the function is decreasing on $\left(-\infty,\frac{1}{2} ight)$. For $x > \frac{1}{2}$, $f'(x) = 12x - 6 > 0$, so the function is increasing on $\left(\frac{1}{2},\infty ight)$. Therefore, the answer is E: decreasing on $\left(-\infty,\frac{1}{2} ight)$; increasing on $\left(\frac{1}{2},\infty ight)$.

Identify the open intervals where the function f(x)=6x2−6x+4 is increasing or \text { Identify the open intervals where the function } f ( x ) = 6 x ^ { 2 } - 6 x + 4 \text { is increasing or } Identify the open intervals where the function f(x)=6x2−6x+4 is increasing or

$\text { Identify the open intervals where the function } f ( x ) = 6 x ^ { 2 } - 6 x + 4 \text { is increasing or }$