91Ó°ÊÓ

Continuity is a fundamental concept in calculus and analysis. It describes a situation where a function does not have any abrupt changes in value. For any point on the curve of a continuous function, you can draw a straight line that just touches the curve at that point without crossing it.

In mathematical terms, a function \( f(x) \) is called continuous on an interval \( [a, b] \) if

• the limit as \( x \) approaches any point \( c \) in that interval exists,
• the function value \( f(c) \) exists, and
• the limit is equal to the function value, i.e., \( \lim_{x \to c} f(x) = f(c) \).

Continuity ensures that no matter how close you get to a point, the function behaves predictably, which is crucial for applying various theorems, such as the generalized Cauchy Mean Value Theorem addressed in the exercise.

Differentiability is related to a function's ability to have a derivative at a given point. A function is differentiable at a point if it has a defined tangent there, and its rate of change can be calculated precisely. Differentiability implies continuity, but the reverse is not always true. A function that is differentiable at every point in an interval \( (a, b) \) is considered differentiable on that interval.

For differentiability, we look for the existence of the derivative:

• If a function's derivative \( f'(x) \) exists, then the function is said to be differentiable at that point.
• The condition is mathematically expressed as:\( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) exists.

Differentiability's predictable behavior of the function allows the application of not only derivatives but also theorems like Cauchy’s Mean Value Theorem, which demands such functions for its applicability.

Rolle's Theorem is a specific case of the mean value theorem and provides useful insights for problems like the one in the original exercise. If you have a continuous function on \( [a, b] \) and it's differentiable on \( (a, b) \), with the value of \( f(a) \) equal to \( f(b) \), Rolle’s Theorem guarantees that there exists at least one point \( c \) in \( (a, b) \) where the derivative \( f'(c) \) equals zero.

This single point represents a local maximum or minimum, depending on the context.

• \( f(a) = f(b) \) ensures a balance across the interval.
• This theorem sets the stage for the generalized results used in more complex calculations in calculus, like being applied in determining a point \( c \) so that certain determinants become zero, very much akin to the problem you are working with.

A determinant is a special number that can be calculated from a square matrix. In the context of the exercise, we utilize a 3x3 matrix determinant which helps in geometrically interpreting the system of equations.

The determinant provides insight into whether a system of equations has a unique solution, no solution, or infinitely many solutions. It’s calculated using a specific formula, often involving permutations of the elements of the matrix.

• For a matrix \( \left| {\begin{array}{ccc}f(a) & g(a) & h(a) \f(b) & g(b) & h(b) \f^{\prime}(c) & g^{\prime}(c) & h^{\prime}(c)\end{array}} \right| \), the determinant is calculated using cross multiplication of these rows and columns.
• A zero determinant, as solved in the exercise, suggests dependencies among the functions or their derivatives used, providing the condition for the existence of such a point \( c \).

Understanding determinants is critical in solving the generalized problem by determining where these functions meet specific criteria or constraints, such as forming a determinant that is zero.