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

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

C

Explanation

Given `P(x)=a_(0)+a_(1)x^(2)+a_(2)x^(4)+………+a_(n)x^(2n)`
where `a_(n)gta_(n-1)gta_(n-2)……..gta_(2)gta_(1)gta_(0)gt0`
`impliesP'(x)=2a_(1)x+4a_(2)x^(3)+……….+2na_(n)x^(2n-1)`
`=2x{a_(1)+2a-(2)x^(2)+…..+na_(n)x^(2n-2)}`………….i
where `(a_(1)+2a_(2)x^(2)+3a_(3)x^(4)+.........+na_(n)x^(2n-2))gt0`
For all `x epsilonR`
Thus, `{(P'(x)gt0, "when" xgt0),(P'(x)gt0, "when"xlt0):}`
i.e. `P'(x)` changes sign from (-ve) to (+ve) at `x=0`
Hence `P(x)` attains minimum at `x=0`
`implies` Only one minimum at `x=0`