Question

Consider the function f : R $\to$ R, defined as $\{\begin{array}{c}{x}^2-x+3,xϵ(-\infty ,3)\cap Q\\ x+a,xϵ(-\infty ,2)-Q\\ 2^x+1,xϵ(2,3)-Q\\ 9tan\left(\frac{\pi x}{12}\right),xϵ[3,6]\end{array}$

If f(x) is continuous at x = 2 then the value of a is

• A. 1
• B. 2
• C. 3
• D. indeterminate

Answer 1

The correct option is C

3

Consider the function f : R $\to$ R, defined as f(x) = $\{\begin{array}{c}{x}^2-x+3,xϵ(-\infty ,3)\cap Q\\ x+a,xϵ(-\infty ,2)-Q\\ 2^x+1,xϵ(2,3)-Q\\ 9tan\left(\frac{\pi x}{12}\right),xϵ[3,6]\end{array}$

$f\left(2^{+}\right)=2^2+1=5$

through irrational

$f\left(2^{-}\right)=2+a$

through rational

f(2) = 4 - 2 + 3 = 5

Hence for continuity at x = 2

2 + a = 5 $\Rightarrow$ a = 3.

$\text{At}x=3\text{For}x=3^{+};f\left(x\right)=9\tan \frac{\pi x}{12}\Rightarrow f\left(3^{+}\right)=9\tan \frac{3\pi }{12}=9f^{\prime }\left(x\right)=9\frac{\pi }{12}{\sec }^2\frac{\pi x}{12}\Rightarrow f^{\prime }\left(3^{+}\right)=9\frac{\pi }{12}{\sec }^2\frac{3\pi }{12}=\frac{3\pi }{2}\approx 4.71\text{For}x=3^{-};f\left(x\right)=\{\begin{array}{cc}{x}^2-x+3;& xϵQ\\ 2^x+1;& xϵR-Q\end{array}\Rightarrow f\left(3^{-}\right)=\{\begin{array}{cc}9;& xϵQ\\ 9;& xϵR-Q\end{array}{f}^{\prime }\left(x\right)=\{\begin{array}{cc}2x-1;& xϵQ\\ 2^x\ln 2;& xϵR-Q\end{array}\Rightarrow f^{\prime }\left(3^{-}\right)=\{\begin{array}{cc}5;& x\in Q\\ 8\ln 2;& x\in R-Q\end{array}\approx \{\begin{array}{cc}5;& xϵQ\\ 5.54;& xϵR-Q\end{array}$
Therefore, continuous but not differentiable at x=3.