How can i do that?

Question

Using logarithmic differentiation, Show that y(x)= xa^2x e^{x^2} has no stationary points other than x=0, if e^-√2 <a <e^√2

Answer 1

lny = lnx + 2xlna + x²; y'/y = 1/x + 2lna + 2x; y' = (1/x + 2lna + 2x)(xa^2xe^{x^2}) = (1 + 2xlna + 2x²)a^2xe^{x^2} = 0

1 + 2xlna + 2x² = 0; D = ln²a - 2 < 0, because e - √2 < a < e^√2; ln(e - √2) < lna < √2.

So there is no x for what y' = 0.

So, this function has no stationary points.