00:01
So for this problem, we're given that b of x is 400x squared minus 2 ,400x cubed, where x is between 0 and 1 .6, and we need to find the maximum value.
00:16
So to find the maximum value, we first need to take the derivative of b of x.
00:20
So the derivative of b of x, if i use my power property, gives me 800x minus 600x.
00:32
7 ,200 x squared.
00:36
Okay, so now in order to maximize my function, i know that that's going to occur when the slope of the tangent line is zero.
00:44
So i'm going to set my derivative equal to zero and solve for x.
00:49
So to start, i'm going to factor out the gcf, which is 800x.
00:55
That leaves me with a 1 minus 9x in parentheses.
00:59
Now i can set each of these terms equal to zero.
01:04
To solve for x.
01:06
So i get x is zero and x equals one -ninth.
01:13
Now i need to decide at which one of these the maximum value occurs.
01:18
So i'm going to guess it's probably not at x equals zero, but let's go ahead and check to make sure it's x equals one -ninth.
01:25
So we know the maximum, we have a maximum value when our derivative is going from increasing to decreasing.
01:37
So i need to plug in a value less than one ninth and see if my function is increasing, and then a value greater than one ninth and see if my function is decreasing.
01:47
If that's the case, i know that i have a maximum value...