Let P x - a 0- a 1 x 2- a 2 x 4- a 3 x 6- a n x 2 n be a polynomial in a real variable x with 0- a 0- a 1- a 2- a n - The function P x hasA- neither a maxima nor a minimaB- only one maximaC- both maxima and minimaD- only one minima

P(x)=a_0+a_1x^2+a_2x^4+a_3x^6+...+a_nx^2n P'(x)=2a_1x+4a_2x^3+6a_3x^5+... ⇒ P'(x)=x(2a_1+4a_2x^2+6a_3x^4+...) If x gt;0, P'(x) is always positive If x lt;0, ...

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Byju's Answer

Standard XII

Mathematics

First Derivative Test for Local Minimum

Let P x = a 0...

Question

Let $P (x) = a_{0} + a_{1} x^{2} + a_{2} x^{4} + a_{3} x^{6} + . . . + a_{n} x^{2 n}$ be a polynomial in a real variable $x$ with $0 < a_{0} < a_{1} < a_{2} . . . < a_{n}$ . The function $P (x)$ has

neither a maxima nor a minima

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only one maxima

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both maxima and minima

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only one minima

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Solution

The correct option is D only one minima
$P (x) = a_{0} + a_{1} x^{2} + a_{2} x^{4} + a_{3} x^{6} + . . . + a_{n} x^{2 n}$
$P^{'} (x) = 2 a_{1} x + 4 a_{2} x^{3} + 6 a_{3} x^{5} + . . .$
$\Rightarrow P^{'} (x) = x (2 a_{1} + 4 a_{2} x^{2} + 6 a_{3} x^{4} + . . .)$
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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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

The correct option is D only one minimaP(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.

Let $P (x) = a_{0} + a_{1} x^{2} + a_{2} x^{4} + a_{3} x^{6} + . . . + a_{n} x^{2 n}$ be a polynomial in a real variable $x$ with $0 < a_{0} < a_{1} < a_{2} . . . < a_{n}$ . The function $P (x)$ has