Question

If $y = 8x^3 - 60x^2 + 144x + 27$ is a strictly decreasing function in the interval:

• A. $(-5, 6)$
• B. $(-\infty, 2)$
• C. $(5, 6)$
• D. $(3, \infty)$
• E. $(2, 3)$

Hint:

For a parabola $ax^2+bx+c$ with $a>0$, the values are negative between the roots. This "Wavy Curve" method allows you to instantly identify the decreasing interval once you find the factors $(x-2)(x-3)$.

Answer 1

Answer: (E)

Concept: A function is strictly decreasing in an interval where its first derivative is strictly less than zero ($y' < 0$). We find the derivative and solve the inequality.

Step 1: Find the derivative $y'$.
Given $y = 8x^3 - 60x^2 + 144x + 27$: \[ y' = 24x^2 - 120x + 144 \]

Step 2: Simplify the derivative and set the inequality.
Factor out the common term 24: \[ y' = 24(x^2 - 5x + 6) \] For strictly decreasing: \[ x^2 - 5x + 6 < 0 \]

Step 3: Solve the quadratic inequality.
Factorizing the quadratic: \[ (x - 2)(x - 3) < 0 \] The roots are $x = 2$ and $x = 3$. Testing the regions:
• $x < 2$: Product is $(-)(-) = +$ (Increasing).
• $2 < x < 3$: Product is $(+)(-) = -$ (Decreasing).
• $x > 3$: Product is $(+)(+) = +$ (Increasing). The function is strictly decreasing in the interval $(2, 3)$.