PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for F (X) = [X] for −1 ≤ X ≤ 1, Where [X] Denotes the Greatest Integer Not Exceeding X Discuss the Applicability of Rolle'S Theorem for the Following Function on the Indicated Intervals ? - PUC Karnataka Science Class 12 - Mathematics

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ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

#### Solution

The given function is $f\left( x \right) = \left[ x \right]$ .

The domain of f is given to be  $\left[ - 1, 1 \right]$.

Let $c \in \left[ - 1, 1 \right]$ such that is not an integer.

Then, $\lim_{x \to c} f\left( x \right) = f\left( c \right)$

Thus, $f\left( x \right)$  is continuous at $x = c$.

Now, let  $c = 0$.

Then,

$\lim_{x \to 0^-} f\left( x \right) = - 1 \neq 0 = f\left( 0 \right)$

Thus,  is discontinuous at = 0.

Therefore,

$f\left( x \right)$  is not continuous in  $\left[ - 1, 1 \right]$ .

Rolle's theorem is not applicable for the given function.

Is there an error in this question or solution?

#### APPEARS IN

Solution F (X) = [X] for −1 ≤ X ≤ 1, Where [X] Denotes the Greatest Integer Not Exceeding X Discuss the Applicability of Rolle'S Theorem for the Following Function on the Indicated Intervals ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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