00:01
In this question, we are asked to find the points where the tangent line to the graph of the function f is horizontal.
00:10
To do that, we need first to find f prime of x.
00:14
And f prime of x by definition equals to the limit of f of x plus h minus f of x over h as h goes to zero.
00:29
Let's calculate f of x plus h.
00:31
To calculate f of x plus h, we simply need to replace x by x, x plus h everywhere this is a formula for f of x plus h now let's simplify it let's distribute let's expand it 2 times x cube plus 3 x squared h plus 3x h squared plus h squared plus 3x h squared plus this is x plus h cube minus 33 times x squared plus 2x h plus h squared plus 144 x plus 144 h now this equals to 2x cube plus 6x squared h plus 6x squared plus 6x squared plus 2h cube minus 33 x squared minus 66 h squared plus 144 x plus 144 h.
02:19
Now let's calculate f of x plus h minus f of x.
02:31
To do that we need to take this huge expression, copy, paste, and this part 2, and subtract f of x from it.
03:06
Minus, and recall that f of x is 2x cube minus 33x squared plus 144x.
03:17
Plus 33x squared minus 144x.
03:25
Now we can do some cancellations.
03:28
We can cancel 2x cube, we can cancel 33x squared, and we can cancel 144x...