Calculus
Polynomial satisfying ODE — continuity condition
MJAT_TS5_P1
Grade 12
Question:
Let $P(x)$ be a polynomial satisfying $P(x)-2P'(x)=3x^3-27x^2+38x+1$. If the function
$$f(x)=\begin{cases}\dfrac{\sin(a-1)\sin(P(x)-2)}{6},+18, & x\neq\dfrac{\pi}{2}\\\dfrac{(ab)\cos\pi}{a+b-3ab}, & x=\dfrac{\pi}{2}\end{cases}$$
is continuous at $x=\dfrac{\pi}{2}$, then $(a+b)$ equals:
Step-by-Step Solution
Key Concept: First find $P(x)$ by solving the ODE $P-2P'=3x^3-27x^2+38x+1$. Assume $P(x)=ax^3+bx^2+cx+d$. Then match coefficients. Evaluate the limit of the function at $x=\pi/2$ and set equal to the value at $x=\pi/2$.
$a+b=\mathbf{2}$.
Correct Answer: 2