Finding Intervals of Increase and Decrease: Analyzing y = 6x² - 15x

To determine the intervals of increase and decrease for the function $y = 6x^2 - 15x$, we begin by finding its first derivative.

The first derivative of the function is found as follows:

$y = 6x^2 - 15x$

$y' = \frac{d}{dx}(6x^2 - 15x) = 12x - 15$

To find critical points, set the derivative equal to zero:

$12x - 15 = 0$

$12x = 15$

$x = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}$

The critical point is $x = 1 \frac{1}{4}$. We need to determine the sign of the derivative on either side of this point to identify the intervals of increase and decrease.

• For $x < 1 \frac{1}{4}$, choose $x = 0$:

$y'(0) = 12(0) - 15 = -15$

Since $y' < 0$, the function is decreasing for $x < 1 \frac{1}{4}$.

• For $x > 1 \frac{1}{4}$, choose $x = 2$:

$y'(2) = 12(2) - 15 = 24 - 15 = 9$

Since $y' > 0$, the function is increasing for $x > 1 \frac{1}{4}$.

Therefore, the function increases for $x > 1 \frac{1}{4}$ and decreases for $x < 1 \frac{1}{4}$.

The correct intervals of increase and decrease for the function $y = 6x^2 - 15x$ are:

$\nearrow: x < 1 \frac{1}{4}$ (Increasing)
$\searrow: x > 1 \frac{1}{4}$ (Decreasing)