91Ó°ÊÓ

A continuous function is one that can be drawn without lifting your pencil from the paper. This means there are no breaks, holes, or jumps in the graph. In mathematical terms, a function \( f(x) \) is continuous on an interval \([a, b]\) if it is continuous at every point in that interval.

Polynomials, like \(f(x) = (x-1)(x-2)(x-3)\), are continuous everywhere because they have no discontinuities. This makes polynomial functions straightforward to work with when checking conditions for theorems like Rolle's.

For Rolle's Theorem, checking that our function is continuous on a closed interval \([1, 3]\) establishes the first requirement. Continuous functions ensure smoothness, vital for transitioning into differentiability.

Differentiability relates to the smoothness of a function's graph. If a function is differentiable at a point, it means it has a well-defined tangent at that point. In simpler terms, you can calculate the slope of the curve precisely.

For polynomial functions, like the one in our exercise, differentiability is guaranteed everywhere. This is because polynomial functions don't have sharp corners or vertical tangents.

Rolle's Theorem requires the function to be differentiable on the open interval \((1, 3)\). Since our function is differentiable everywhere, this condition is easily met. Differentiability ensures we can apply calculus tools like derivatives, which are needed in finding where the slope of the function equals zero.

Polynomial functions are algebraic expressions that involve sums of powers of variables with coefficients. In general, a polynomial function can be written as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0\). Here, \(a_n, a_{n-1}, ..., a_0\) are coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial.

The function given in our exercise, \(f(x) = (x-1)(x-2)(x-3)\), is a cubic polynomial. Cubic polynomials have one or more bends in their graphs.

Polynomial functions are not only continuous and differentiable but also easy to handle computationally. This makes them ideal candidates for applying theorems like Rolle's. They allow us to explore solutions and roots simply and effectively.

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\). The solutions to this equation are called the roots and can be found using various methods, such as factoring, completing the square, or the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

In our solution, we derived a quadratic equation from the derivative: \(3c^2 - 12c + 11 = 0\). Solving this equation gives us the critical points where the derivative is zero. These points are where the function's slope changes direction, an essential aspect of understanding the behavior of the function on the interval.

Solving the quadratic equation can sometimes tell us about the unique or multiple points satisfying certain conditions, like in Rolle's Theorem, where the slope of the tangent is zero within a certain range.