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) = -12x⁵ + 75x⁴ - 80x³

Accepted Answer

Welcome back, everyone. Determine the intervals on which the function G of X equals 3x the power of 4 minus 16 x cubed plus 24 x quad is increasing and decreasing. If we want to identify the monotonicity intervals of a function, we have to first of all differentiate it. So let's identify the derivative. So we're taking the derivative of 3 X to the power of 4th. We are subtracting the derivative of 16 X cubed. And then we are adding the derivative of 24 X2, and we can apply the power rule. So we're going to get 3 multiplied by 4, that's 12 X cubed. Minus 16 multiplied by 3 deaths 48 X squared. Plus 24 multiplied by 2, that's 48 X, right? Each exponent is decreased by 1 unit according to the Power rule and now we can factor it out. We can factor out 12 X, which leaves us with X2 minus 4 X. Plus 4 and we can factor it out further because inside of parentheses we have a perfect square that's x minus 2 squared. Now let's recall that the function is increasing when the derivative is positive, and it is decreasing when the derivative is negative. So we want to identify those intervals. And what we can do is simply plot a number line, right, because we need to identify the critical points. If we set this derivative equal to 0, we will have two critical points. One of them is 0 because 12 X can be set to 0 to get the overall product of 0, and this gives us X of 0. And the second critical point is X minus 2 equals 0, so that means X is equal to 2. So let's show that we have set G of X equals to 0, and this gives us our critical points of X equals 0 and X equals 2. And now, considering the number line, we want to identify those intervals where the derivative is positive and negative. So for example, we can take our first point, which is less than 0, and so for let's say G at -1. So this gives us 12 multiplied by -1, multiplied by -1 minus 22. Now, do we need to check this math? Well, not really, because, well, we have two factors. So, Essentially, we can check if this number is negative or positive. However, the easiest way is to check if 12 x is positive. Well, if X is negative, then 12 x is negative, right? So the first part is negative if X is less than 0, and the second part is never negative because it's a square of a number, right? So the first factor is negative for X being less than 0, but the second factor is positive. So negative multiplied by positive gives us an overall negative sign. Now between 0 and 2, well, we can take a value such as 1, so 12 X will be positive. And the square of a number, well, it's never negative, right? So it's going to be positive as well. Then we can take any value greater than 2, so 12 x is going to be positive, and X minus 2 is going to be positive as well. It's never negative. So positive multiplied by positive gives us the positive sign for the derivative. Positive multiplied by positive gives us the overall positive sign for the derivative. So now let's understand that. The function G. Of a Is increasing. When X belongs to the interval from 0 to infinity, right? Non-inclusive. Because, well, specifically at 0, we might have a critical, we, we actually have a critical point, right? So the derivative is exactly 0, it's not positive. And then we're combining two onto our interval because it's a saddle point. The derivative doesn't change its sign, so we can just say that the function keeps increasing past point X equals 2, so we're just combining our two intervals into one interval where G is. Increasing. So GFX is increasing from 0 up to infinity, right? Because this is where we have the positive sign for the derivative, and then our function GFX is decreasing. If X belongs to the interval from negative infinity. Up to 0. And those will be our answers. 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) = -12x⁵ + 75x⁴ - 80x³

Key Concepts

Critical Points

First Derivative Test

Intervals of Increase and Decrease

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