The correct interval where the function is strictly increasing is
x>−6/5. So, the correct choice would be D. (-2 < x < ∞).
How to determine where the function is only increasing
To determine where the function f(x) = x² + 9x² + 24x + 15 is only increasing, we can analyze its derivative.
Given
f(x) = x² + 9x² + 24x + 15, let's find its derivative:
f'(x) = d/dx (x² + 9x² + 24x + 15)
f'(x) = 2x + 18x + 24
f'(x) = 20x + 24
For the function to be strictly increasing, its derivative must be positive throughout the interval in question. To find when the derivative is positive, we set it greater than zero:
20x +24 > 0
Solving for x:
20x>−24
x > -24/20
x> -6/5
This indicates that the function is strictly increasing when x> -6/5
Among the options provided:
A. -∞ < x < ∞ (This interval includes all x values and doesn't specify where the function is strictly increasing.)
B. -∞ < x < -2 (This interval includes x values less than -6/5.)
C. -5 < x < -2 (This interval includes x values less than -6/5.)
D. -2 < x < ∞ (This interval includes x values greater than -6/5.)
The correct interval where the function is strictly increasing is
x>−8/9
. So, the correct choice would be D. (-2 < x < ∞).