Let f(x)=1+22x2+3x2x4+42x6....+n2x2n-2=+(n+1)2x2n, then f(x) has:
exactly one minimum
exactly one maximum
at least one maximum
None of these
Determine the condition of f(x)
Given that:
f(x)=1+22x2+3x2x4+42x6....+n2x2n-2=+(n+1)2x2n
Differentiating the given equation:
f'(x)=2.22x+4.3x2x3+6.42x5....+(2n-2).n2x2n-3=x(2.22+4.3x2x2+6.42x4....+(2n-2).n2x2n-4)
Now, for maxima or minima,
f'(x)=0⇒x=0
Therefore, f(x)) has only minimum which is 0.
Hence, the correct answer is Option (A).