Determining the Interval of Increasing Growth Rate | A Derivative Analysis of a Function

Let f(x)=15/1 +4e^−0.2x . Over what interval is the growth rate of the function increasing?

To find the interval over which the growth rate of a function is increasing, we need to find the derivative of the function and determine where it is positive

To find the interval over which the growth rate of a function is increasing, we need to find the derivative of the function and determine where it is positive.

Given the function f(x) = 15/(1 + 4e^(-0.2x)), let’s find its derivative.

Using the quotient rule, the derivative of f(x) is:

f'(x) = [(1 + 4e^(-0.2x))(0) – 15(4e^(-0.2x)(-0.2))] / (1 + 4e^(-0.2x))^2

Simplifying further:

f'(x) = (12e^(-0.2x)) / (1 + 4e^(-0.2x))^2

To determine where the growth rate is increasing, we need to find where the derivative f'(x) is positive.

Setting f'(x) > 0 and solving for x:

(12e^(-0.2x)) / (1 + 4e^(-0.2x))^2 > 0

To make the inequality simpler, we can multiply both sides by (1 + 4e^(-0.2x))^2, noting that this expression is always positive:

12e^(-0.2x) > 0

Since e^(-0.2x) is always positive for any value of x, we can divide both sides of the inequality by e^(-0.2x) without changing the sign:

12 > 0

This inequality holds true for all real numbers, so the growth rate of the function f(x) = 15/(1 + 4e^(-0.2x)) is increasing over the entire real number line, or over the interval (-∞, +∞).

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