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

##### 1 Answer

The function is convex for

#### Explanation:

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

Setting this equal to zero, we know that the

Therefore, we end up with

i.e. we have one degree of

If you try to find some rational roots, it turns out that none of the candidates

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

The new problem is that the original function has these breaks at

I:

II:

III:

IV:

V:

We know that the actual second derivative is

We can see that only terms that will determine the sign of the function are the two polynomials since

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

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