# How do you find the points of inflection of the curve #y=e^(x^2)#?

##### 2 Answers

None, it's concave up for

#### Explanation:

Compute the first derivative.

#y' = 2xe^(x^2)#

Second derivative is given by the product rule.

#y'' = 2(e^(x^2)) + 2x(2x)e^(x^2)#

#y'' = 2e^(x^2) + 4x^2e^(x^2)#

We need to set this to

#0 = 2e^(x^2) + 4x^2e^(x^2)#

#0 = 2e^(x^2)(1 + 2x^2)#

We see this has no solution because

This doesn't have a real value so no real solution to this equation. This simply means the function

Hopefully this helps!

In principle, by differentiating twice, setting the result to zero, and checking whether the result is a genuine point of inflexion. However, this function has no points of inflexion.

#### Explanation:

Differentiating twice gives

This can never be zero because all three terms of the product are always strictly positive

Did you mean

You should then test that the second derivative changes sign at these points, which it clearly does as