Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
-$$f ( x ) = - 2 x ^ { 2 } + 3 x + 1$$
A) Increasing on $( - \infty , 1 )$; decreasing on $( - 1 , \infty )$
B) Increasing on $( - \infty , 0 )$; decreasing on $( 0 , \infty )$
C) Increasing on $\left( - \infty , \frac { 3 } { 4 } \right)$; decreasing on $\left( \frac { 3 } { 4 } , \infty \right)$
D) Increasing on $\left( - \infty , \frac { 3 } { 8 } \right)$; decreasing on $\left( \frac { 3 } { 8 } , \infty \right)$

Question

Quizplus · Accepted Answer

To determine the intervals of increase and decrease, we find the derivative of the function $f(x) = -2x^2 + 3x + 1$, which is $f'(x) = -4x + 3$. Setting $f'(x) = 0$ gives $x = \frac{3}{4}$. Since $f'(x)$ changes sign from positive to negative at $x = \frac{3}{4}$, the function is increasing on $\left( - \infty , \frac { 3 } { 4 } \right)$ and decreasing on $\left( \frac { 3 } { 4 } , \infty \right)$.

Find the Intervals on Which the Function Is Increasing and the Intervals