Maximum possible number of extrema of the function?

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?

Solution 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.

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