Question

Consider the following statements :
1. The function is continuous at every point in the interval $(1,\infty )$.
2. The function is differentiable at x = 1.5. Which of the above statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is B 2 only

To check continuity of function, the function should be continuous throughout it's domain
$For1\le x\le 2,f\left(x\right)=2-x$ which is a polynomial function and always continuous throughout it's domain

$\Rightarrow$ The function is differentiable at $x=1.5$

For $x>2,f\left(x\right)=3x-x^2$ which is also a a polynomial function and continuous and differentiable throughout it's domain

For continuity , the left hand side and right hand side limits should be equal at the point where functions change

Now, $RHL=\underset{x\to 2^{-}}{lim}(2-x)=0$ and

$LHL=\underset{x\to 2^{+}}{lim}(3x-x^2)=2$

$LHL\ne RHL$, hence the function isn't continuous at every point in the interval $(1,\infty )$