Maximum possible number of extrema of the function?

Question

Consider a function :

$$ f(x)= P(x)e^{-(x^4+2x^2)} $$in the domain $x \in (-\infty,\infty)$, $P(x)$ is any polynomial of degree $k$. What is the maximum possible number of extrema of the function.

My attempt : I differentiated the function and finally got a polynomial of degree $k+3$ set to zero ( condition for extremum). Therefore, I conclude that maximum possible extrema should be $k+3$ corresponding to k+3 roots of the polynomial which i get after differentiating. But the answer given in the book is $k+1$. Please help me understand this? What am i missing?

Answer 1

If you differentiate it then you will get a term in multiplication $4x^3 +4x =4x(x^2+1)$ now $x^2 +1$ has no real roots so effectively differentiation will be zero for only $k+1$ times.

Answer 2

Before differentiating assume that the polynomial Pk(x) is of zero-order i.e k=0 (P(x)=constant). Now upon differentiating it with respect to x and equating it to zero, you'll only get one term i.e. P(k=0)(x).(4x^3+4x).e^(f(x)) = 0. Now as e^(f(x)) can't be zero. Hence P(k=0)(x).(4x^3+4x)=0. Upon simplifying 4x(x^2+1)=0 as x^2+1=0 doesn't have any real solution. So x=0. Now from this you can interpret this that for a zero-order polynomial you are getting one extrema. Hence for "k" order of polynomial you'll get "k+1" extrema.