Finding the Constant in an Anti-Derivative Function

[NDiggity]

Here is the question:

Find a function f which satisfies both of the following properties:
f ' (x) = x^3
The line x + y = 0 is tangent to the graph of f.

I figured out that f(x) is 1/4x^4 + C. Now I don't know what to do. I know I need to figure out C but I'm stuck. I isolated x+y=0 for y to get y= -x, and the derivative of that is -1, so the slope of the tangent line is -1. So I then figured out what x value causes x^3 to also be -1, and it turns out to be -1. This is the x-coordinate at which the line is tangent to. So the y coordinate would be -1 + y=0. So y is 1. The point at which the line is tangent to f is (-1,1). If everything up to this point is correct, how do I find C?

 

[EnumaElish]

What equation does/should the general solution satisfy at (-1,1)?

 

[NDiggity]

Do I go 1=1/4(-1)^4 + C and solve for C?

 

[HallsofIvy]

Yes. The important thing is understanding why you can do that. In order that the graph of your function be tangent to x+ y= 0 at x= -1, the graph has to pass through (-1,1). That is true only if 1= (1/4) (-1)^4+ C.

 

[NDiggity]

Thank you so much for explaining that to me, now I can go write my Math 110 final...yay?