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Answer :
To determine the intervals on which the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex] is decreasing, we need to follow these steps:
1. Find the Derivative:
- The first step is to find the derivative of the function, [tex]\( f(x) \)[/tex], which we'll call [tex]\( f'(x) \)[/tex]. The derivative tells us the rate at which [tex]\( f(x) \)[/tex] is changing and is essential to finding where the function is increasing or decreasing.
2. Solve for Critical Points:
- Set the derivative [tex]\( f'(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex]. These values of [tex]\( x \)[/tex] are called critical points. They are important because a function can change from increasing to decreasing (or vice versa) at these points.
- For the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex], the critical points are [tex]\( x = -6 \)[/tex] and [tex]\( x = 0 \)[/tex].
3. Determine Intervals of Increase and Decrease:
- Use the critical points to divide the real number line into intervals.
- Choose test points from each interval and substitute them into the derivative [tex]\( f'(x) \)[/tex] to determine the sign of [tex]\( f'(x) \)[/tex] within each interval.
- If [tex]\( f'(x) \)[/tex] is negative on an interval, the function is decreasing on that interval.
4. Identify Decreasing Intervals:
- After testing the intervals around the critical points, we observe whether the function is increasing or decreasing.
- In the case of the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex], there are no intervals where the derivative is negative between the critical points. Therefore, the function is not decreasing on any interval.
Based on these steps, there are no intervals on the real number line where the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex] is decreasing.
1. Find the Derivative:
- The first step is to find the derivative of the function, [tex]\( f(x) \)[/tex], which we'll call [tex]\( f'(x) \)[/tex]. The derivative tells us the rate at which [tex]\( f(x) \)[/tex] is changing and is essential to finding where the function is increasing or decreasing.
2. Solve for Critical Points:
- Set the derivative [tex]\( f'(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex]. These values of [tex]\( x \)[/tex] are called critical points. They are important because a function can change from increasing to decreasing (or vice versa) at these points.
- For the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex], the critical points are [tex]\( x = -6 \)[/tex] and [tex]\( x = 0 \)[/tex].
3. Determine Intervals of Increase and Decrease:
- Use the critical points to divide the real number line into intervals.
- Choose test points from each interval and substitute them into the derivative [tex]\( f'(x) \)[/tex] to determine the sign of [tex]\( f'(x) \)[/tex] within each interval.
- If [tex]\( f'(x) \)[/tex] is negative on an interval, the function is decreasing on that interval.
4. Identify Decreasing Intervals:
- After testing the intervals around the critical points, we observe whether the function is increasing or decreasing.
- In the case of the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex], there are no intervals where the derivative is negative between the critical points. Therefore, the function is not decreasing on any interval.
Based on these steps, there are no intervals on the real number line where the function [tex]\( f(x) = 2x^4 + 16x^3 - 21 \)[/tex] is decreasing.
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