We want to know where this function is. Increasing, decreasing It is a polynomial function. So when we examine the critical points, we know that it can't be non differential. So we really just want to look for critical points where the derivative is zero. The derivative will involve the power rule derivative of two X to the fifth would be 10 x to the fourth for the derivative of 15 x of the fourth over four. I'm gonna bring that for in front and subtract one from its power. So the four that I bring in front actually cancels with the denominator, and then the power drops to the third power and then the last term. If I bring the three in front, it will cancel with the Dominator, and my new power is just X squared. Critical points will be where this is equal to zero. So let's set 10 acts of the fourth minus 15 x cubed plus five x squared equal to zero. We can pull a common five x squared outfront, which would give me two X squared minus three x plus one. Okay, we knew a little trial and air on the two x squared minus three x plus one. But that will factor 22 acts minus one times X minus one. Now, any time that we have that equal to zero, we've got a critical point. Well, five X squared is equal to zero when X equals zero X minus one is equal to zero when X equals one for the two X minus one. If I set that equal to zero, it's really just a two step equation. I have one divide by the two, so I get an answer of X equals 1/2. Okay, so now we're going to analyze this. We're gonna put this on a number line and check the sign of our derivative be of X equals zero X equals 1/2 or 0.5 X equals one. Now we will pick points on all sides. Let's start with the X value of negative one, and we're filling this into F prime. If I feel that in for F prime, I get 10 plus 15 plus five, definitely a positive. So we're looking at an increasing function now. We need to be ah, little bit careful with these regions that are coming up because we have them as decimals. Let's try filling in 0.25 for the value between 0.5. So I'm gonna go 10 times, 100.25 to the fourth, minus 15 times 150.25 to the third, plus five times 50.25 square. That gives me a really small, positive answer. I get 0.117 so it's still increasing for a value between 0.5 and one. I'm going to use 10.75 if I type that in 10 times 0.75 to the fourth, minus 15 times 150.75 cubed plus five times 50.75 squared. I get a slightly negative answer, so that's a decreasing region. Finally, if I fill in two 10 times, two to the fourth, minus 15 times two cubed plus five times to square in answer of 60. So that is a positive derivatives, so it increases, so there's actually two increasing regions. This whole region here is considered increasing. Zero is increasing because it didn't ever change to a downward trajectory. So we're looking at an increasing function, from negative infinity to 1/2 as well as one to infinity. The Onley decreasing region is between the X values of 1/2 and one