A function f is defined for all real numbers and has the...

A function $f$ is defined for all real numbers and has the following three properties: $$ f(1)=5 \quad f(3)=21 \quad f(a+b)-f(a)=k a b+2 b^{2} $$ for all real numbers $a$ and $b$ where $k$ is a fixed real number independent of $a$ and $b$. (a) Use $a=1$ and $b=2$ to find $k$. (b) Find $f^{\prime}(3)$. (c) Find $f^{\prime}(x)$ for all real $x$.

A function $f$ is defined for all real numbers and has the following three properties:
$$
f(1)=5 \quad f(3)=21 \quad f(a+b)-f(a)=k a b+2 b^{2}
$$
for all real numbers $a$ and $b$ where $k$ is a fixed real number independent of $a$ and $b$.
(a) Use $a=1$ and $b=2$ to find $k$.
(b) Find $f^{\prime}(3)$.
(c) Find $f^{\prime}(x)$ for all real $x$.

Key Concepts

Polynomial Functions and Differentiation

When a functional equation suggests that the differences of a function are given by a polynomial expression, it often implies that the function itself is a polynomial. Recognizing that a function is polynomial simplifies the process of differentiation since the derivative can be computed term by term using standard rules.

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This concept underpins many methods of finding rates of change and is essential when differentiating functions that are defined through difference relations or functional equations.

Functional Equations

A functional equation is an equation where the function appears in terms of its values at various inputs, rather than just its independent variable. It constrains the function in a way that must hold for all inputs, and such equations often reveal the underlying structure or form of the function.

Substitution in Functional Equations

Substitution involves choosing specific values for the variables in a functional equation to simplify the expression and solve for unknown parameters. By plugging in particular choices, one can isolate constants or determine function values which are critical in solving or identifying the function.

Transcript

00:01 The total function obeys these properties here.

00:05 So that f of 1 equals 5, f of 3 equals 21, f of a plus b equal minus f of a equals k times a, k times a times b plus 2b squared.

00:20 So i'm not sure what, you know, why the general, if this has any general applications, but okay, we have some function like that.

00:30 And so they tell us to use a equals 1 and b equals 2 to find what the derivative of this function is at 3, and also to figure out what the derivative for any value of x.

00:47 So we can kind of see here that this is x, if this is x and this is x, so a is x, and this would be h.

00:54 And so what we're seeing here is that we have this thing here looks like a linear function.

01:00 Right? so this would be actually a, would be a quadratic function.

01:07 So i think it must be quadratic, since this would be linear in x.

01:13 So if it was quadratic, then we'd have to have four.

01:21 Yeah, it's something i think it'd have to be quadratic.

01:27 So we'd have maybe five.

01:31 We could figure out what it was.

01:33 Well, no, we couldn't, just from these two points, we couldn't figure out what it is.

01:38 We'd have an arbitrary constant left over.

01:42 Because a quadratic, you need three points.

01:44 Oops, not it...