Maxima and MinimaMedium
Question
If P(x) = a0 + a1x2 + a2x4 + ....... + anx2n be a polynomial in x ∈ R with 0 < a1 < a2 < .... < an, then P(x) has-
Options
A.no point of minimum
B.only one point of minimum
C.only two points of minimum
D.None of these
Solution
We have P(x) = a0 + a1 x2 + a2 x4 + ..... + anx2n
For max. or min. P′(x) = 0
⇒ 2x[a1 + 3a2 x2 + ...... + n. anx2n-2] = 0
⇒ x = 0
Now P″(x) = 2a1 + 12a2x2 + .... + 2n(2n-1) anx2n-2
⇒ P″(x) = 2a1 > 0
Hence, P(x) has only one minimum at x = 0.
For max. or min. P′(x) = 0
⇒ 2x[a1 + 3a2 x2 + ...... + n. anx2n-2] = 0
⇒ x = 0
Now P″(x) = 2a1 + 12a2x2 + .... + 2n(2n-1) anx2n-2
⇒ P″(x) = 2a1 > 0
Hence, P(x) has only one minimum at x = 0.
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