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

Solution :
The given polynomial is
$$P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ..... + {a_n}{x^{2n}},\,x\, \in \,R{\text{ and }}\,0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.$$
Here, we observe that all coefficients of different powers of $$x,$$ i.e., $${a_0},\,{a_1},\,{a_2},\,.....,{a_n},$$     are positive.
Also, only even powers of $$x$$ are involved.
Therefore, $$P\left( x \right)$$  cannot have any maximum value.
Moreover, $$P\left( x \right)$$  is minimum, when $$x = 0,$$  i.e., $${a_0}.$$
. Therefore, $$P\left( x \right)$$  has only one minimum.