(1 point) The function
f(x)=4x3+24x2−252x+1
is decreasing on the interval ________________
Enter your answer using the interval notation for open intervals.
It is increasing on the interval(s) __________________
The function has a local maximum at _________________
To determine whether a function is decreasing or increasing on a certain interval, we need to analyze its derivative. The derivative of a function gives us information about its rate of change.
1. To find the derivative, we can use the power rule of differentiation. For each term in the function, we take the derivative of the power and multiply it by the coefficient:
f'(x) = 12x^2 + 48x - 252
2. Next, we need to find the critical points of the derivative. These are the values of x where the derivative equals zero or does not exist. In this case, we have a quadratic equation, and we can solve it for x:
12x^2 + 48x - 252 = 0
Using factoring or the quadratic formula, we find that x = -4 and x = 7 as the critical points.
3. Now, we create a number line and mark the critical points along with any other relevant points. Our critical points are x = -4 and x = 7.
-----(-inf)---(-4)---(7)---(inf)-----
4. Finally, we test a value from each interval on the derivative to determine whether it's positive or negative.
For x < -4:
Let's substitute x = -5 into the derivative:
f'(-5) = 12(-5)^2 + 48(-5) - 252 = 0 + (-240) - 252 = -492
Since the result is negative, the derivative is negative, indicating that the function is decreasing on the interval (-inf, -4) (open interval notation).
For -4 < x < 7:
Let's substitute x = 0 into the derivative:
f'(0) = 12(0)^2 + 48(0) - 252 = -252
Since the result is negative, the derivative is negative, indicating that the function is decreasing on the interval (-4, 7) (open interval notation).
For x > 7:
Let's substitute x = 8 into the derivative:
f'(8) = 12(8)^2 + 48(8) - 252 = 768 + 384 - 252 = 900
Since the result is positive, the derivative is positive, indicating that the function is increasing on the interval (7, inf) (open interval notation).
Therefore, the function f(x) is decreasing on the interval (-inf, -4) and (-4, 7), and it is increasing on the interval (7, inf).
As for the local maximum, it occurs at the critical point x = -4.