Let $P\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+\cdots +a_n{x}^{2n}$be a polynomial in a real variable x with $0<a_0<a_1<a_2<\cdots <a_n$.The function P(x) has

The given polynomial is

$p\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+\dots +a_n{x}^{2n},x\in R$

and $0<a_0<a_1<a_2<\dots <a_n$

Here, we observe that all coefficients of different powers of x,

i.e., a₀,a₁,a₂…,a_n, are Positive.

Also, only even powers of x are involved.
Therefore, P(x) cannot have any maximum value.
Moreover, P(x) is minimum, when x=0, i.e., a₀
Therefore, P(x) has only one minimum.

Alternative method:

We have

$\begin{array}{r}{P}^{\mathrm{\prime }}\left(x\right)=2a_{1}x+4a_2{x}^3+\dots +2n{a}_n{x}^{2n-1}\\ =x\left(2a_1+4a_2{x}^2+\dots +2n{a}_n{x}^{2n-2}\right)\end{array}$

Clearly, $P\text{'}\left(x\right)> 0 for x > 0 and P\text{'}\left(x\right)< 0 for x < 0.$
Thus, P(x) increases for all x > 0 and decreases for all x < 0.
Therefore, $P\text{'}\left(x\right) has x=0$ as the point of manima.