00:01
In this question, we are asked to find the x -coordinates of all local minima of the function g.
00:07
And we will first calculate g ' of x to find the critical points.
00:12
G ' equals to 11x to the 10 minus 27x to the 8th power.
00:19
We can rewrite that as x to the 8th multiplied by 11x squared minus 27.
00:28
Now, we want that to be equal to 0.
00:31
We want to solve the equation x to the 8th multiplied by 11x squared minus 27 equals 0.
00:40
That splits into two equations.
00:41
X to the 8th equals 0.
00:44
And the second case is 11x squared minus 27 equals 0.
00:51
From the first equation, x equals 0.
00:54
From the second, 11x squared equals to 27.
01:00
That means that x squared equals to 27 over 11, which means that x is plus minus the square root of 27 over 11.
01:17
That gives us three critical points.
01:24
The critical points are x equals 0, x equals to the square root of 27 over 11, and x equals negative square root of 27 over 11.
01:39
Now, what we will do next is we will calculate the second derivative of the function g.
01:46
We will calculate g' ' and to do that we need to differentiate g'.
01:51
So g' ' of x equals to 110x to the 9 minus 27 multiplied by 8x to the 7.
02:20
That's going to be x to the 7 multiplied by 110x squared minus 216...