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Calculus Level 4

A polynomial P ( x ) P(x) with real coefficients has the property that P ′ ′ ( x ) ≠ 0 P''(x) \ne 0 for all x x . If P ( 0 ) = 1 P(0)=1 and P ′ ( 0 ) = − 1 P'(0)=-1 , what can you say about P ( 1 ) P(1) ?

None of the rest P ( 1 ) ≤ 0 P(1)\le 0 P ( 1 ) ≥ 0 P(1)\ge 0 − 1 2 < P ( 1 ) < 1 2 \frac { -1 }{ 2 } <P(1)<\frac { 1 }{ 2 }

Mehul Chaturvedi by Mehul Chaturvedi , 22, India

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1 solution

Chew-Seong Cheong
Jan 20, 2015

P ′ ′ ( x ) ≠ 0 P''(x) \ne 0 for all x ⟹ P ′ ′ ( x ) = a x \implies P''(x) = a , where a a is a constant. Therefore, P ( x ) P(x) is of the form P ( x ) = a x 2 + b x + c P(x) = ax^2 + bx + c .

Then, we have

P ( 0 ) = 1 a ( 0 ) + b ( 0 ) + c = 1 ⟹ c = 1 \begin{aligned} P(0) & = 1 \\ a(0)+b(0)+ c & = 1 \\ \implies c& = 1 \end{aligned}

Also

P ′ ( 0 ) = − 1 2 a ( 0 ) + b = − 1 ⟹ b = − 1 \begin{aligned} P'(0) & = -1 \\ 2a(0) + b & = -1 \\ \implies b & = -1 \end{aligned}

Therefore, P ( x ) = a x 2 − x + 1 ⟹ P ( 1 ) = a P(x) = ax^2 - x + 1 \implies P(1) = a which is an unknown. Hence the answer is None of the rest \boxed{\text{None of the rest}} .

4 Helpful 0 Interesting 0 Brilliant 0 Confused

I did it in the same way.But my polynomial was of degree n

Sayan Das - 4 years, 7 months ago

does "for all x" include unreal values for x or only real values?

Maadhav Gupta - 4 years, 5 months ago

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All values including complex ones.

Chew-Seong Cheong - 4 years, 5 months ago

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