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Let P(x)=a0+a1x^2+a2x^4++an x^(2n) be a ...

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

neither a maximum nor a minimum

B

only one maximum

C

only one minimum

D

only one maximum and only one minimum

Text Solution

Verified by Experts

The correct Answer is:
C
Given, ` P(x) =a _0 + a_1 x ^(2) + a_2 x ^(4) + … + a _n x ^( 2n)`
where, ` a _n gt a_(n- 1) gt a_(n-2) gt … gt a _2 gt a_1 gt a_0 gt0`
`rArr P' (x) = 2a_1 x + 4 a_2 x^(3) + … + 2na_n x ^( 2 n - 2)`
`" " = 2x ( a_1 + 2 a _2 x^(2) + ... + n a_n x ^( 2n - 2 )`
where, ` (a _1+ 2 a_2 x ^(2) + 3 a _ 3 x ^(4) + ... + n a _n x^(2n - 2) ) gt 0, AA x in R`.
Thus, ` " " {{:(P'(x) gt 0",",, "when " x gt 0 ), (P'(x) lt 0 "," ,, " where " x lt 0):}`
i.e. `P'(x)` changes sign from (- ve) to ( + ve) at ` x=0`
`therefore P(x)` attains minimum at ` x=0`
Hence, it has only one minimum at ` x = 0`.
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