Question

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

Polynomials

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

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
$$b+12=3\Rightarrow b=−9.$$
Now putting $$c=\frac{3}{2}$$ and $$b= -9\ in\ {eq1}$$ we get

$$a−36+48=5\Rightarrow a=−7.$$

So required second degree polynomial is $$P(x)=−7−9x+(\frac{3}{2})x2$$