Let `P(x)=a_0+a_1x^2+a_2x^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(x)` has a. neither a maximum nor a minimum b. only one maximum c. only one minimum d. only one maximum and only one minimum e. none of these

Options

A. neither a maximum nor a minimum
B. only one maximum
C. only one minimum
D. only one maximum and only one minimum

Answer

3

Explanation

The given polynomial is
`P(x) =a_(0)x^(2)+a_(2)x^(4)…+a_(n)x^(2n),x in R`
and `0lta_(0)lta_(1)lta_(2)lt..lta_(n)`
Here we observe that all coefficients of different powers of x ltrbgt i.e j`a_(0),a_(1),a_(2)..a_(n)` are positve
Also only even power of x are involved
Therefore P(x) connot have any maximum value
Moreover P(x) connot have any maximum value
Moreover P(x) is minimum when x =0 i.e `a_(0)`
Therefore P(x) has only one minimum
Alternative method:
We have
P(x) =`2a_(1)x+4a_(2)x^(3)+..2na_(n)x^(2-1)`