To tackle the question regarding a surjective differentiable function and its properties, let's break down the concepts involved and analyze the conditions given. We need to find the least integral value of a certain variable, which we will denote as $n$, such that there exists some function $f$ that meets the criteria for all functions in consideration.

Understanding Surjectivity and Differentiability

A function is said to be **surjective** (or onto) if every element in the codomain has a pre-image in the domain. In simpler terms, for a function $f:A\to B$, for every $b\in B$, there exists at least one $a\in A$ such that $f(a)=b$.

When we talk about a function being **differentiable**, it means that the function has a derivative at every point in its domain. A function that is **twice differentiable** has a first derivative that is also differentiable, which means we can find the second derivative.

Finding the Least Integral Value

Given that we are looking for the least integral value of $n$ such that there exists some function $f$ satisfying the conditions of being surjective and twice differentiable, we can consider the implications of these properties.

• **Surjectivity** implies that the function must cover all values in its range, which can be influenced by the nature of the function itself.
• **Twice differentiable** means that the function's curvature can change, allowing for more complex behaviors that can help achieve surjectivity.

To find $n$, we can consider polynomial functions, as they are smooth (infinitely differentiable) and can be manipulated to achieve surjectivity. A polynomial of degree $n$ can have at most $n$ roots, but it can also be designed to cover all real numbers if $n$ is odd. For instance, a cubic function (degree 3) can be made surjective by ensuring it has appropriate leading coefficients.

Example of a Surjective Function

Consider the cubic function $f(x)=x^3$. This function is:

• **Differentiable** everywhere, with $f^{\prime }(x)=3x^2$, which is always non-negative.
• **Surjective** because as $x$ approaches $\mathrm{\infty }$ or $-\mathrm{\infty }$, $f(x)$ covers all real numbers.

For a function to be surjective and twice differentiable, we can conclude that a polynomial of degree 3 is sufficient. Therefore, the least integral value of $n$ that satisfies the conditions of the problem is:

Final Result

The least integral value of $n$ is 3.

This means that for any function that is surjective and twice differentiable, we can find a polynomial of degree 3 that meets these criteria. This example illustrates how the properties of differentiability and surjectivity can be intertwined through the choice of function.