91Ó°ÊÓ

Exponential functions are fundamental in mathematics and play a significant role in various fields like calculus, physics, and economics. An exponential function can be expressed in the form \( f(x) = e^{g(x)} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Exponential functions exhibit unique properties like an initial value set at \( x=0 \), known as the y-intercept which, for exponential functions, is usually \( f(0) = 1 \) if no transformations are applied.

• Growth or decay is characterized by the exponent \( g(x) \).
• If \( g(x) \) is linear, the function grows or decays at a constant rate.
• Their graphs are non-linear, typically J-shaped, representing exponential growth or an inverted J representing decay.

In the context of functional equations like the one given in the problem, the structure \( f(x+y+z) = f(x)f(y)f(z) \) is suggestive of multiplication of probabilities or repeated compounded growth, hence the connection to an exponential format.

Differential calculus is a branch of mathematics that focuses on how functions change when their inputs change. The key idea is differentiation, which finds the derivative—a measure of how a function changes as its input changes.
The derivative of a function \( f(x) \) with respect to \( x \), denoted \( f'(x) \), describes the rate at which \( f(x) \) is changing at any particular point.

• If the derivative is positive, the function is increasing.
• If it is negative, the function is decreasing.
• When the derivative is zero, the function is constant or is at a local maximum/minimum.

When evaluating specific points like \( f'(0) = 3 \), it implies that at \( x = 0 \), the function \( f(x) \) is changing at a rate of 3 units per unit change in \( x \).

The Role of Differentiation in the Exercise

In our exercise, we differentiated the function \( f(x) \) hypothesized as \( e^{g(x)} \). We then used the condition \( f'(0) = 3 \) to derive a constant rate of change for \( g(x) \), and subsequently confirmed that \( f'(2) = 12 \) using this derivative format.

Cauchy's functional equation is an equation of the form \( g(x + y) = g(x) + g(y) \). This equation is critical in understanding linearity within functions.
The general solution to Cauchy's equation, when \( g \) is continuous, is \( g(x) = cx \) for some constant \( c \). This linear form provides the basis for many results in functional equations.
In this exercise, this form was extended to \( g(x + y + z) = g(x) + g(y) + g(z) \), reinforcing that the underlying function \( g \) is indeed linear provided the function is differentiable everywhere.

Application of Cauchy's Equation in the Exercise

By assuming the structure \( f(x) = e^{g(x)} \), and recognizing that \( f(x+y+z) = f(x)f(y)f(z) \) leads to the expanded equation \( g(x+y+z) = g(x) + g(y) + g(z) \), we see the application of Cauchy’s equation in deducing that \( g(x) \) must be linear and continuous, hence \( g(x) = 3x \) given \( g'(0) = 3 \). This foundation helped us determine the value of \( f'(2) \) accurately.