Let the function f be defined as f(x)={ (P(x))/(x-2), x ≠2 7, x=2 . where P(x) is a po

step 1 In this question we have to find PX. So first we will see three times differentiation of PX equa

Let the function f be defined as f(x)={ (P(x))/(x-2), x...

Let the function $f$ be defined as $f(x)=\left\{\begin{aligned} \frac{P(x)}{x-2}, & x \neq 2 \\ 7, & x=2 \end{aligned}\right.$ where $P(x)$ is a polynomial such that $P^{\prime \prime \prime}(x)$ is identically equal to 0 and $P(3)=9 .$ If $f(x)$ is continuous at $x=2$, then (A) $P(x)=2 x^{2}-x-6$ (B) $P(x)=2 x^{2}+x-6$ (C) $P(x)=2 x^{2}-x+6$ (D) None of these

Let the function $f$ be defined as $f(x)=\left\{\begin{aligned} \frac{P(x)}{x-2}, & x \neq 2 \\ 7, & x=2 \end{aligned}\right.$
where $P(x)$ is a polynomial such that $P^{\prime \prime \prime}(x)$ is identically equal to 0 and $P(3)=9 .$ If $f(x)$ is continuous at $x=2$, then
(A) $P(x)=2 x^{2}-x-6$
(B) $P(x)=2 x^{2}+x-6$
(C) $P(x)=2 x^{2}-x+6$
(D) None of these

Key Concepts

Factorization in Limit Evaluation

When evaluating limits that result in an indeterminate form such as 0/0, factorization is a key tool. By factoring the numerator (or denominator) to cancel common factors, the limit can be resolved, which is particularly useful in ensuring continuity at points where the original expression was undefined.

Continuity of Piecewise Functions

Continuity at a point means that the limit of a function as the variable approaches that point equals the function's value at the point. In piecewise-defined functions, this concept is crucial because different formulas define the function on different parts of the domain. Ensuring continuity may involve setting the limit of one piece equal to the explicitly given value at a specific point.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined or is defined differently at a point due to a factor that can be canceled in the expression. This usually happens when the numerator and denominator of a rational function share a common factor. By canceling this factor and redefining the function appropriately at that point, the discontinuity can be removed, making the function continuous.

Polynomial Characterization from Derivative Conditions

If a polynomial’s higher order derivative (for example, the third derivative) is identically zero, it implies that the polynomial is of a degree less than that order. In this context, a zero third derivative indicates that the polynomial is quadratic or of lower degree, which greatly restricts its general form and aids in determining its coefficients.

Transcript

Please give Ace some feedback