Here is the question:

The lines x + y= 0 and x + y= 4/3 are tangent to the graph of y = F(x) where F is an antiderivative of -x^2. Find F.

The answer is F(x)= 1/3(2-x^3)

I'm just not too sure how to get that answer!

Thanks for all your help!! :)

Interesting question. Well, since F is an antiderivative of -x^2 then we know that F must be of the form

F(x) = -(1/3)x^3 + C, C = constant

where I simply integrated -x^2. Now, the problem is to determine C. Well, we know that the two lines are tangent to F(x), meaning they both intersect F but at different (x,y) values. Let's put the two lines into slope-intercept form:

1) y = -x
2) y = -x + 4/3

Okay so it is clear now that both lines have a slope of -1. We should next find out where our function F has a slope of -1. Simply take the derivative of F which brings us back to F'(x) = -x^2. Note that F'(x) represents the slope. So to find out where the slope is -1, simply sub -1 in for F'(x) and solve for x:

-1 = -x^2
=> x^2 = 1
=> x = +/- 1

Hence our function F(x) has a slope of -1 at x = 1 and x = -1.

Now here we use the fact that lines (1) and (2) intersect F(x) where the slope is -1. The problem is... does line (1) intersect at -1 and line 2 at +1, or vice versa? To determine this, we will try both.

(***) Assume first that (1) intersects F at x=1. Then we can find the y value by subbing x=1 into eqn (1):

y = -x => y = -(1) => y = -1

So the point is (1,-1). We then know that this point also lies on y = F(x) = -(1/3)x^3 + C, so sub this point into F to find C:

(-1) = -(1/3)(1)^3 + C
=> C = -2/3

Is this the correct C value? Well to check, we now utilize equation number (2). Recall we assumed (1) intersected F at x = 1, so then (2) must intersect F at x = -1. Sub x = -1 into (2):

y = -x + 4/3 => y = -(-1) + 4/3 => y = 7/3

So then (x,y) = (-1, 7/3) must also lie on F. Let's check:

y = F(x) = -(1/3)x^3 - 2/3
=> 7/3 = -(1/3)(-1)^3 - 2/3
=> 7/3 = 1/3 - 2/3
=> 7/3 = -1/3

Oh oh! What happened?! Well, go back to (***) where we ASSUMED that (1) intersects F at x=1. Obviously, this was the wrong assumption. The only other option is that (1) intersects F at x = -1. This should give us the answer we're looking for. So simply repeat the procedure starting from (***):

Sub x = -1 into (1):

y = -x => y = -(-1) => y = 1

So the point (-1,1) must also lie on F. Sub this point into F:

y = F(x) = -(1/3)x^3 + C (recall our old C value was garbage)
=> 1 = -(1/3)(-1)^3 + C
=> 1 = 1/3 + C
=> C = 2/3

Hence the CORRECT C value is 2/3 and so our final solution for F(x) is

F(x) = -(1/3)x^3 + 2/3

or we can factor out -1/3 to get the answer you stated:

-----------------------------------
F(x) = 1/3 (2 - x^3)
-----------------------------------

We can verify this is correct by again noting that since (1) intercepts F at x = -1 then (2) must intercept F at x = +1. Sub x = 1 into (2):

y = -x + 4/3 => y = -(1) + 4/3 => y = 1/3

So we MUST also have (x,y) = (1, 1/3) lying on F. Let's check:

y = F(x) = -(1/3)x^3 + 2/3
=> 1/3 = -(1/3)(1)^3 + 2/3
=> 1/3 = 1/3

much better.

Alright well there ya go.