00:01
First, we take a derivative to this function, which gives us 1 fourth times 4 times x to a power of 4 minus 1, minus 1 3 times 3 times x to a power of 3 minus 1, minus 2 x to a power of 2 minus 1, which is x to a power of 3 minus 1, which is x to a power of 3 minus 2.
00:34
To minus 2x.
00:39
Part a, we want to find the point on the graph of f where the slope of the tangent line is equal to minus 2x.
00:52
And hence, it tells us the derivative function equals to minus 2x, which also substitutes the derivative function with its expression and we simplify this expression.
01:32
And we can see that it has two solutions.
01:36
X equals 0 or x equals 1.
01:47
And when the function value of f0 as 0, the function value of f1 equals 1 fourth times 1, minus 1 third times 1 minus 1, which is 1 4th minus 1 3rd minus 1, which is 12.
02:17
The denominator is 12, and the numerator is 3 minus 4 minus 12, which equals minus 13 over 12.
02:35
And hence this 2 .0, 0, and 1, minus 13 over 2.
02:45
For part b, the derivative of this function equals 0, which gives us x squared minus x, sorry, x cubed minus x squared minus 2x, equals 0.
03:08
And we can rewrite this as x times x squared minus x minus 2 equals 0.
03:19
It's x plus 1, x minus 2.
03:41
And it gives us 3 solutions...