JEE Mathematics 2025 Question with Solution

Given: $f$ is differentiable on $\mathbb{R}$, $f(2)=4$, and

Find: The number of points where the curve

meets the x-axis.

Take natural logarithm of the given limit form. Since $f(2)=4$,

Hence,

Now write

as $x \to 0$. Therefore, for the exponential limit to be finite, we use the standard derivative form and get

The solution is internally inconsistent: it states $f(2)=1$ in one place, which contradicts the question text $f(2)=4$, and the subsequent working is not reliable. However, the solution explicitly concludes that the correct option is A.

Using the solution as the primary answer authority, the correct option is A, which corresponds to the curve meeting the x-axis $2$ times.

The extracted solution contains multiple errors:

1. It replaces the given condition $f(2)=4$ by $f(2)=1$.
2. It informally treats $\alpha$ as if it were a function value expression.
3. It claims a cubic must have exactly $2$ real roots, which is not a valid general statement.
4. The second approach rewrites the given limit incorrectly.

Because of these contradictions, the algebraic derivation in the solution cannot be trusted as valid working. Still, the solution explicitly marks Option A as correct, so the extracted answer is A.