Let the function f be defined on the set of real numbers by f(x) = \begin{cases}x^2 - x, & if x < 1\\\\\frac{x^2 - 1}{3}, &if x \geq 1\end{cases} then which of the following statement is **TRUE **?

Question 21

Let the function $$f$$ be defined on the set of real numbers by

$$ f(x) = \begin{cases}x^2 - x, & if x < 1\\\frac{x^2 - 1}{3}, &if x \geq 1\end{cases}$$ then which of the following statement is TRUE ?

Solution

$$f'\left(x\right)=2x-1\ \ ,\ x<1$$

$$f'\left(x\right)=\frac{2x}{3}\ \ ,\ x\ge\ 1$$

$$f'\left(1^+\right)=\frac{2}{3}\ \ \ and\ f'\left(1^-\right)=1$$ => Right hand Derivative is not equal to LHD. Hence is is not differentiable at x=1

Left hand limit = 0 and RHL = 0. Hence it is continuos at x=1

Create a FREE account and get: