For what values of x is f(x)= e^x/(x^2-x) -e^xf(x)=exx2xex concave or convex?

1 Answer
Jul 18, 2018

The function is convex for x < -1.660 x<1.660, 0 < x < 10<x<1, x > 1.928x>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)=exx2xex
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.