00:01
In this question, we are asked to find the critical points of the function, the intervals of increase and decrease, and the local maxima and minima.
00:08
To find the critical points, we need to calculate y ' y ' equals to 1 minus 1 over x to the fourth multiplied by 4x cubed.
00:19
I used the chain rule.
00:21
By the chain rule, i first differentiated ln, and the derivative of ln is 1 over x to the fourth, and then multiplied by the derivative of the expression inside the ln, which is 4x cubed.
00:32
X cubed cancels, and we will get 1 minus 4 over x, and this equals to x minus 4 divided by x.
00:43
Now we want to find the critical points.
00:46
We want to find the points where the denominator, where the numerator is 0, and that happens when x equals 4, and when the, where the denominator is 0.
00:57
However, note that the function is not defined at x equals 0, because ln 0 is not defined, right? therefore, we will cross it out.
01:07
So the only critical point is x equals 4.
01:15
Now we will draw the number line and put x equals 4 onto the number line, and then that point, x equals 4, splits the number line into two subintervals, from negative infinity to 4, and from 4 to infinity.
01:35
Now we need to study the sign of f of y ' on each of the subintervals...