How do you find the points of inflection of the curve #y=e^(x^2)#?
None, it's concave up for
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.
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