Consider the function f(x)=casesP(x)(x-2), & x27, & x=2cases, where P(x) is a polynomial s…

Question

Consider the function $f(x)=\begin{cases}\dfrac{P(x)}{\sin(x-2)}, & x
e2\7, & x=2\end{cases}$, where $P(x)$ is a polynomial such that $P'''(x)$ is always a constant and $P(3)=9$. If $f(x)$ is continuous at $x=2$, then $P(5)$ is equal to ...........

Answer: 39

Accepted Answer

$P'''$ constant means $P$ is degree 2: $P(x)=(x-2)(ax+b)$. Continuity at $x=2$: $\displaystyle\lim_{x	o2}\frac{(x-2)(ax+b)}{\sin(x-2)}=2a+b=7$ ... (i)

$P(3)=1\cdot(3a+b)=9\Rightarrow3a+b=9$ ... (ii)

Subtracting: $a=2,\ b=3$. So $P(x)=(x-2)(2x+3)$, and $P(5)=3	imes13=39$.

Solutions · 1