Find a second-degree polynomial P such that P(4)=5, P'(4)=3, and P"(4)=3

Find a second-degree polynomial P such that P(4)=5, P'(4)=3, and P"(4)=3

Question
Polynomials
asked 2021-02-02
Find a second-degree polynomial P such that P(4)=5, P'(4)=3, and P"(4)=3

Answers (1)

2021-02-03
Consider second degree polynomial \(P(x)=a+bx+cx^{2}\) such that
\(P(4)=5, P′(4)=3\), and \(P′′ (4)=3.\)
\(P(4)=5P(4)=5\) gives us
\(a+4b+16c=5\).
Now differentiating P, we get \(P′=b+2cx\). Now putting x=4, we get
\(b+8c=3.
Now differentiating P′, we get \(P''=2cP\). Now putting x=4, we get
\(b+12=3\Rightarrowb=−9.\)
Now putting \(c=\frac{3}{2} and \(b= -9 in {eq1}\) we get
\(a−36+48=5\Rightarrowa=−7.\)
So required second degree polynomial is \(P(x)=−7−9x+(\frac{3}{2})x2\)
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