Find a second-degree polynomial P such that P(2)=5, P'(2)=3, and P''(2)=2

Tammy Fisher

Tammy Fisher

Answered question

2021-11-09

Find a second-degree polynomial P such that P(2)=5, P'(2)=3, and P''(2)=2

Answer & Explanation

Todd Williams

Todd Williams

Beginner2021-11-10Added 18 answers

Let the second degree polynomial is
P(x)=ax2+bx+c
Differentiate with respect to x
P(x)=(ax2+bx+c)
Using the sum rule, we can write
P(x)=(ax2)+(bx)+(c)
Use the power rule to differentiate
P(x)=2ax21+bx11+0
P'(x)=2ax+b
Differentiate again
P''(x)=(2ax)'+(b)'
P''(x)=2a+0
P''(x)=2a
Substitute x=2 in P''(x)=2a, to get
P''(2)=2a
It is given that P''(2)=2
2=2a
Divide both sides by 2
1=a
Substitute x=2 and a=1 in P'(x)=2ax+b, to get
P'(2)=2(1)(2)+b
It is given that P'(2)=3
3=4+b
Subtract 4 from both sides
-1=b
Substitute x=2, a=1 and b=-1 in P(x)=ax2+bx+c, to get
P(2)=(1)(2)2+(1)(2)+c
It is given that P(2)=5
5=4-2+c
5=2+c
Subtract 2 from both sides
3=c
Substitute the values of a, b, c in the original equation, to get
P(x)=x2x+3
Results:
P(x)=x2x+3

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