JEE Mathematics 2024 Question with Solution

Given: $f(x) = |2x^2 + 5|x| - 3|$, $x \in \mathbb{R}$

Find: The value of $m+n$, where $m$ is the number of points of discontinuity and $n$ is the number of points of non-differentiability.

The solution states that the function is a composition of polynomials and absolute value functions. Such functions are continuous everywhere on $\mathbb{R}$.

Therefore,

For differentiability, the solution identifies possible non-differentiable points from the modulus terms.

The outer expression is non-differentiable where

Solving this gives the critical points

Also, the inner term $|x|$ is non-differentiable at $x=0$, which is already included among these points. Hence the total number of non-differentiable points is

Thus,

Therefore, the correct option is D.

Given: $f(x) = |2x^2 + 5|x| - 3|$

Find: Number of discontinuity points and non-differentiability points.

From the extracted solution, continuity is checked first. Since polynomial expressions and modulus functions are continuous, their composition remains continuous for every real value of $x$.

Hence,

Next, non-differentiability occurs where a modulus changes sign or where an inner modulus term is not differentiable.

Set the inside of the outer modulus equal to zero:

The provided solution concludes that this gives three critical real points:

Therefore, the number of points where $f$ is not differentiable is

Finally,

So the required answer is $3$, that is, option D.