Calculus question from Howard Anton

Question

Let $y=f(x)$ and $dy/dx=e^y+2y+x$. I wish to prove that the critical points are only relative minima.

I am stuck with this problem from the book which needs expert help. I have no idea how to proceed.

Answer 1

We are given $dy/dx = Z+x$ where $Z$ involves $y$.

When $dy/dx = 0$ AND $d^2y/dx^2$ is Negative , we have Critical Maxima Point.
When $dy/dx = 0$ AND $d^2y/dx^2$ is Positive , we have Critical Minima Point.

We can see that $d^2y/dx^2 = W dy/dx + 1$ , where $W$ corresponds to $Z$ & $1$ corresponds to $x$.
Here , when-ever $dy/dx = 0$ , we get $d^2y/dx^2 = W \times 0 + 1 = 1$ , which is Positive & not Negative.
Hence , at all the Critical Points , $d^2y/dx^2$ Positive indicates relative Minima.