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Question

Let P(x)=a0+a1x2+a2x4+a3x6+...+anx2n be a polynomial in a real variable x with 0<a0<a1<a2...<an. The function P(x) has
  1. neither a maxima nor a minima
  2. only one maxima
  3. both maxima and minima
  4. only one minima


Solution

The correct option is D only one minima
P(x)=a0+a1x2+a2x4+a3x6+...+anx2n
P(x)=2a1x+4a2x3+6a3x5+...
P(x)=x(2a1+4a2x2+6a3x4+...)
If x>0,P(x) is always positive
If x<0,P(x) is always negative
Hence, P(x) has only one minima.

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