JEE Advanced · Mathematics · 23. C&D

Let $f : (0, 1) \to ℝ$ be the function defined as $f (x) = [4 x] {(x - \frac{1}{4})}^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following is(are) true?

A The function $f$ is discontinuous exactly at one point in $(0, 1)$
B There is exactly one point in $(0, 1)$ at which the function $f$ is continuous but NOT differentiable
C The function $f$ is NOT differentiable at more than three points in $(0, 1)$
D The minimum value of the function $f$ is $- \frac{1}{512}$

Verified Solution

Answer & Solution

Correct Answer

(B) There is exactly one point in $(0, 1)$ at which the function $f$ is continuous but NOT differentiable

Step-by-step Solution

Detailed explanation

Given,

f : (0, 1) \to ℝ

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let f:0,1→ℝ be the function defined as fx=4xx-142x-12, where [x] denotes the greatest integer less than or equal to x. Then which of the following is(are) true?

Step-by-step Solution

Let $f : (0, 1) \to ℝ$ be the function defined as $f (x) = [4 x] {(x - \frac{1}{4})}^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following is(are) true?