Determining the Function that Increases at the Fastest Rate | A Comparative Analysis of Derivatives

Which function increases at the fastest rate between x = 0 and x = 8?

To determine which function increases at the fastest rate between x = 0 and x = 8, we can compare the derivatives of the functions

To determine which function increases at the fastest rate between x = 0 and x = 8, we can compare the derivatives of the functions.

The derivative of a function represents its rate of change at any given point. If the derivative is positive, the function is increasing; if it is negative, the function is decreasing.

First, let’s consider two different functions: f(x) and g(x).

To find the function that increases at the fastest rate, we need to compare the derivatives of f(x) and g(x) in the interval x = 0 to x = 8.

Let’s say f'(x) represents the derivative of f(x), and g'(x) represents the derivative of g(x).

We need to analyze whether f'(x) or g'(x) is bigger within the interval x = 0 to x = 8.

To do this:

1. Differentiate both functions, finding f'(x) and g'(x). This will give us the slope of the functions at any given point.

2. Evaluate f'(x) and g'(x) at the endpoints of the interval (x = 0 and x = 8).

3. Compare the values of f'(0) and g'(0), as well as f'(8) and g'(8).

The function that has the larger derivative at the endpoints of the interval will be increasing at the fastest rate.

For example, let’s consider two functions:
f(x) = 3x^2 + 2x – 1
g(x) = 4x^3 + 2x^2 – 3x

To find f'(x):
f'(x) = 6x + 2

To find g'(x):
g'(x) = 12x^2 + 4x – 3

Next, let’s evaluate the derivatives at the endpoints of the interval:

Let’s find f'(0):
f'(0) = 6(0) + 2 = 2

Let’s find g'(0):
g'(0) = 12(0)^2 + 4(0) – 3 = -3

Now, let’s find f'(8):
f'(8) = 6(8) + 2 = 50

Let’s find g'(8):
g'(8) = 12(8)^2 + 4(8) – 3 = 779

Comparing the values:
f'(0) = 2, g'(0) = -3
f'(8) = 50, g'(8) = 779

From the comparison, we can conclude that the function g(x) = 4x^3 + 2x^2 – 3x increases at the fastest rate between x = 0 and x = 8, as its derivative g'(x) has larger values at both endpoints of the interval.

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