Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8

Accepted Answer

Welcome back everyone to another video. Determine the intervals on which the function F of X equals X to the power of 4 divided by 4 minus 5 X cubed plus 28X2 + 68 is increasing and decreasing. So first of all, let's analyze where the function is increasing. F of X is increasing. If its derivative is positive, so we want to identify F of X, and we want to identify the intervals where the derivative is positive. So now what we're going to do is basically differentiate F of X, we're going to apply the power rule. So X, the power of 4th gives us the derivative of 4 X cubed, 4 divided by 4 gives us 1, so we're getting X cubed minus 5. multiplied by 3, which is 15 X squad plus 2 multiplied by 28, that's 56 X. So we have successfully applied the power rule. The derivative of 68 is 0, right? And now what we want to do is simply factor it out. So, first of all, let's factor out X. This gives us X squared minus 15 X plus 56, which can be factored out further. So, that's X multiplied by X minus 7, multiplied by X minus 8. Now, what we're going to do is simply plot a number line and identify the critical values, right? So, essentially, we want to identify the zeros of this function. The first one is 0, right? So we're going to add 0 because it's setting X to 0 gives us the overall product of 0, considering the second factor X equals 7, which gives us another 0. Followed by x equals 8, so we have 3 possible zeros, and this gives us a total of 4 intervals to consider from negative infinity up to 0, from 0 to 7, from 7 to 8, and from 8 up to positive infinity. So now let's consider the sign of each factor. Starting with X, if we take any X value, which is less than 0, X is going to be negative. X minus 7 is going to be negative as well, right? X minus 8 is going to be negative as well. Let's say we take -1, -1 minus 8 gives us -9. So when we multiply 3 negative values, we're going to get negative multiplied by negative, which is positive. Multiplied by negative, this gives us the overall negative product. Now, let's continue with any x value between 0 and 7. Let's say 1, so X would be positive, 1 minus 7 is negative, 1 minus 8 is negative. So positive multiplied by negative, multiplied by negative gives us the overall positive product. Between 7 and 8, X is positive. X minus 7 is positive. Let's say we take 7.5, subtract 7. This gives us 0.5, which is positive, and 7.5 minus 8. Well, this gives us a negative factor. So now in this case, our overall product is negative, positive multiplied by positive, multiplied by negative. And then any value greater than 8 gives us X, X minus 7, and X minus 8. So once again, the overall product here is positive, and we can see that if we're interested in where the function is increasing, we have to take those positive intervals. So FFX is increasing. When x belongs to two intervals, so from 0 to 7, not inclusive because 0 and 7 are critical points. Where the function attains its minimum or maximum, right? So our first interval is from 0 to 7, non-inclusive. And So we can say union from 8 up to infinity. Then our function F of X is decreasing when the first derivative is negative, so X belongs to two intervals from negative infinity. Up to 0. And from 7 to 8. And that's it. We have obtained our final answers. Let's stay with them and thank you for watching.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8

Key Concepts

Critical Points

First Derivative Test

Intervals of Increase and Decrease

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