00:01
For this problem, we want to find the value of k in the function y, which is equal to negative 2x squared plus kx plus 5.
00:09
If y equals 4x plus 13 is a line tangent to this curve.
00:17
Note that in here, k is less than zero or negative.
00:21
Now if y equals 4x plus 13 is tangent to this curve, then this, this.
00:30
The slope of the tangent line, let's call it m sub t, is equal to the derivative of this curve, y prime.
00:40
Now, y prime is equal to negative 4x plus k.
00:48
And the slope of this tangent line is the coefficient of x, which is 4.
00:53
So if m sub t equals y prime, then negative 4x plus k equals 4 implies that k is equal to 4x plus 4.
01:12
Also, if a line is tangent to a certain curve, there must be one point wherein they cross or where they intersect.
01:21
That means if y equals 4x plus 13, we can replace the y in this curve by 4x plus 13.
01:31
So the line and the curve will intersect when 4x plus 13 equals negative 2x squared plus kx plus 5.
01:42
So if k equals 4x plus 4, then 4x plus 13 equals negative 2x squared plus 5.
01:52
Plus 4x plus 4 times x plus 5.
01:57
So 4x plus 13 equals negative 2x squared plus 4x squared plus 4x plus 5.
02:06
That means you can cancel out this 4x.
02:10
And we have 8 equal to 2x squared...