Question

For what values of x is `f(x)= e^x/(x^2-x) -e^x` concave or convex?

Answer 1

The function is convex for `x < -1.660`, `0 < x < 1`, `x > 1.928`

Explanation:

We need to find points of inflection for this function, i.e. where the second derivative is 0.

`f(x) = e^x/(x^2-x) - e^x`
`f'(x) = (e^x(x^2-x) - e^x(2x-1))/(x^2-x)^2 - e^x`
`f''(x) = ((x^2-x)^2[e^x(x^2-3x+1) + e^x(2x-3) ])/(x^2-x)^4 - (e^x(x^2-3x+1)* 2 (x^2-x)(2x-1))/(x^2-x)^4 - e^x`

Setting this equal to zero, we know that the `e^x` will never go to zero, so we can divide that out. Similarly, we can multiply by the `(x^2-x)^4` in the denominator by specifying that `x ne 0, 1`.

Therefore, we end up with
`0 = (x^2-x)^2[x^2-3x+1 + 2x-3] - 2(x^2-3x+1)(x^2-x)(2x-1) - (x^2-x)^4`
`0 = (x^2-x)[(x^2-x)(x^2-x-2) - 2(x^2-3x+1)(2x-1) - (x^2-x)^3]`
i.e. we have one degree of `x^2-x` which doesn't cancel out the 4 we multiplied, so we can forget about it as well. Now we have to expand out this sixth degree polynomial:

`0 = x(x-1)(x^2-x-2) - 2(x^2-3x+1)(2x-1) - x^3(x-1)^3`
`0 = -x^6 + 3x^5 -2x^4 -5 x^3 + 13x^2 - 8x + 2`
`0 = x^6 - 3x^5 + 2x^4 + 5x^3 - 13x^2 + 8x - 2`

If you try to find some rational roots, it turns out that none of the candidates `(pm1, pm2)` work, hence the roots are all irrational.
We can plot this function to see:

graph{x^6-3x^5+2x^4+5x^3-13x^2+8x-2 [-3, 3, -10, 10]}

And see there are two real solutions at around
`x_1 = -1.66 and x_2 = 1.928`

The new problem is that the original function has these breaks at `x = 0, 1` so the regions are
I: `x < x_1`
II: `x_1 < x < 0`
III: `0 < x < 1`
IV: `1 < x < x_2`
V: `x_2 < x`

We know that the actual second derivative is
`f''(x) = (-e^x(x^6 - 3x^5 + 2x^4 + 5x^3 - 13x^2 + 8x-2))/(x^2-x)^3`

We can see that only terms that will determine the sign of the function are the two polynomials since `e^x > 0`

From the above graph, we see that in region I and V, the long polynomial is positive and regions II-IV it is negative.

We can also see that `x^2 - x` is negative only in region III.

This means that the regions have alternating concavities, starting with convex.