Answer :
To determine the intervals on which the function [tex]\( f(x) = x^4 - 4x^3 - 19 \)[/tex] is increasing, we follow these steps:
1. Find the derivative of the function.
The first step is to differentiate the function [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]. The derivative, [tex]\( f'(x) \)[/tex], helps to determine where the function is increasing or decreasing.
[tex]\[
f'(x) = \frac{d}{dx}(x^4 - 4x^3 - 19) = 4x^3 - 12x^2
\][/tex]
2. Find the critical points.
Critical points occur where the derivative [tex]\( f'(x) = 0 \)[/tex] or is undefined. Since [tex]\( f'(x) = 4x^3 - 12x^2 \)[/tex], we set it equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
4x^3 - 12x^2 = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
4x^2(x - 3) = 0
\][/tex]
This gives the critical points [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].
3. Test intervals around the critical points.
To determine whether the function is increasing or decreasing in the intervals defined by the critical points, we test points in each interval:
- Interval [tex]\( (-\infty, 0) \)[/tex]:
Choose a test point, like [tex]\( x = -1 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(-1) = 4(-1)^3 - 12(-1)^2 = -4 - 12 = -16
\][/tex]
Since [tex]\( f'(-1) < 0 \)[/tex], the function is decreasing in the interval [tex]\( (-\infty, 0) \)[/tex].
- Interval [tex]\( (0, 3) \)[/tex]:
Choose a test point, like [tex]\( x = 1 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(1) = 4(1)^3 - 12(1)^2 = 4 - 12 = -8
\][/tex]
Since [tex]\( f'(1) < 0 \)[/tex], the function is decreasing in the interval [tex]\( (0, 3) \)[/tex].
- Interval [tex]\( (3, \infty) \)[/tex]:
Choose a test point, like [tex]\( x = 4 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(4) = 4(4)^3 - 12(4)^2 = 256 - 192 = 64
\][/tex]
Since [tex]\( f'(4) > 0 \)[/tex], the function is increasing in the interval [tex]\( (3, \infty) \)[/tex].
4. Conclusion on intervals where [tex]\( f(x) \)[/tex] is increasing:
After testing the intervals, we find that the function [tex]\( f(x) = x^4 - 4x^3 - 19 \)[/tex] is increasing on the interval [tex]\( (3, \infty) \)[/tex].
1. Find the derivative of the function.
The first step is to differentiate the function [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]. The derivative, [tex]\( f'(x) \)[/tex], helps to determine where the function is increasing or decreasing.
[tex]\[
f'(x) = \frac{d}{dx}(x^4 - 4x^3 - 19) = 4x^3 - 12x^2
\][/tex]
2. Find the critical points.
Critical points occur where the derivative [tex]\( f'(x) = 0 \)[/tex] or is undefined. Since [tex]\( f'(x) = 4x^3 - 12x^2 \)[/tex], we set it equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
4x^3 - 12x^2 = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
4x^2(x - 3) = 0
\][/tex]
This gives the critical points [tex]\( x = 0 \)[/tex] and [tex]\( x = 3 \)[/tex].
3. Test intervals around the critical points.
To determine whether the function is increasing or decreasing in the intervals defined by the critical points, we test points in each interval:
- Interval [tex]\( (-\infty, 0) \)[/tex]:
Choose a test point, like [tex]\( x = -1 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(-1) = 4(-1)^3 - 12(-1)^2 = -4 - 12 = -16
\][/tex]
Since [tex]\( f'(-1) < 0 \)[/tex], the function is decreasing in the interval [tex]\( (-\infty, 0) \)[/tex].
- Interval [tex]\( (0, 3) \)[/tex]:
Choose a test point, like [tex]\( x = 1 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(1) = 4(1)^3 - 12(1)^2 = 4 - 12 = -8
\][/tex]
Since [tex]\( f'(1) < 0 \)[/tex], the function is decreasing in the interval [tex]\( (0, 3) \)[/tex].
- Interval [tex]\( (3, \infty) \)[/tex]:
Choose a test point, like [tex]\( x = 4 \)[/tex]. Substituting into [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(4) = 4(4)^3 - 12(4)^2 = 256 - 192 = 64
\][/tex]
Since [tex]\( f'(4) > 0 \)[/tex], the function is increasing in the interval [tex]\( (3, \infty) \)[/tex].
4. Conclusion on intervals where [tex]\( f(x) \)[/tex] is increasing:
After testing the intervals, we find that the function [tex]\( f(x) = x^4 - 4x^3 - 19 \)[/tex] is increasing on the interval [tex]\( (3, \infty) \)[/tex].
