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) = -2x⁴ + x² + 10

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the intervals on which the function G of X equals -8X4 plus X2 + 17 is increasing and decreasing. A says the function is increasing on the interval negative infinity quarter and decreasing on a quarter infinity. B says it's increasing on negative infinity negative quarter and decreasing on negative or quarter infinity. C says it's increasing on the intervals negative infinity 25 and 0 quarter and decreasing on the intervals negative 25, 0 and 25 infinity, while D says it's decreasing on negative infinity, 25, 025, 25, 0, and quarter infinity. Now, to figure out where our function is increasing and decreasing, the first thing we need to do is to differentiate our function, OK? So differentiating geopics. OK. Then it's derivative Is going to be equal to -32 X cubed, the derivative of -8X4 plus 2X the derivative of X2 plus 0, the derivative of our constant. Now we differentiate geoX because at the critical points of our function. The derivative or its derivative is going to be equal to 0. So we can set the derivative to zero and solve for the values of X to find the X values of the critical points, and that will help us to analyze our intervals. So at this point. This tells us that negative 32 X cubed plus 2 X will be equal to 0. If we factor 02 X, that's going to be -16 X2 + 1. 2 X multiplied by -16X2 + 1 which is 0 and no if that's the case, this means that 2 X will be equal to 0. And -16 X2 + 1 will be equal to 0. In those scenarios, if 2 X is 0, then X must be 0, and if negative 16 X2 + 1 is 0, that means X2 will be 1/16 and thus X will be equal to + R minus 1/4, OK? So, therefore, X equals 0 and X equals plus or minus quarter are the critical points of our function G of X. Now, why are our critical points important? Well, let's try to visualize them on a, on the X-axis, OK, for our X points. Now if you think about it here, so we have 0, 25, and negative infinity to the left, and then 25% and infinity to the right. And now, if you think about it here, we have 4 intervals. One from negative infinity to 25, 1 from 25 to 01 from 0 to 25, and one from 25 to infinity. Now, if the first derivative is positive for the X value at that point, OK, or with for the X values within that interval, that means our function is increasing. And if the first derivative is negative for the X values in that interval, that means our function is decreasing. So we can pick a point in each of these intervals, look at the nature of the first derivative of X, and that will help us to figure out whether it's in the interval is increasing or decreasing. So let's start to choose our, let's start to analyze our intervals. Let's start where X is in the interval negative infinity, negative quarter. We're gonna pick a value of X and substitute it into the derivative. Let's say we want to use the value when X equals -1. Then our first derivative. It's gonna be equal to -32 multiplied by -1 cube. Plus 2 multiplied by -1. That equals 32 minus 2, which equals 30, 30 is positive. So this tells us that when X equals -1, GRX is. Increasing Because the first derivative is positive. Let's do that for the next set of intervals. Next, when X is in the interval from quarter 0, let's use the value when X equals -1A, OK? In that case, then our first derivative. Would be equal to -32. Multiplied by 1/8 cubed. Plus 2 multiplied by -18. That would be equal to a 1/16 minus 1/4. Which would be equal to -316. Now our first derivative is negative, OK? So this tells us that our function. Is going to be decreasing, OK. So it's decreasing for the interval -250. Now, what about we X? Be in the interval from 0 to 1.25. Well, no, we can use the value when X equals 18. In that case, the first derivative. It's gonna be equal to -32 multiplied by 1/8 cubed plus 2 multiplied by 1/8. Which is gonna be -1. OK, -116. Plus a quarter, yeah, which would be equal to 3/16 because this value is positive, it tells us that our function will be increasing. On this interval. Finally, When X is in the interval a quarter infinity, OK, we can use the value when X E equals 1. Now, the first derivative. Will be equal to -32 multiplied by 1 cubed plus 2 multiplied by 1. That's -32 + 2, which equals -30. Negative So our function is. Decreasing. So what can we tell here? Well, This basically tells us that GeoffX, let me try to fit everything here. This means GeoffX is increasing on negative infinity, quarter, and 0 quarter. It's increasing on both those intervals, while it's decreasing on the intervals negative quarter 0 and a quarter infinity. If we look back at our answer choices, that tells us then that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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

f(x) = -2x⁴ + x² + 10

Key Concepts

Critical Points

First Derivative Test

Intervals of Increase and Decrease

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