When a function does not have any critical numbers, it typically means that there are no points at which the function’s derivative is either zero or undefined. Critical numbers are important because they often indicate potential local maxima, minima, or inflection points in a function’s graph. Let’s break this down further using the function f(x) = 5x^3 – x^2 + 25 as an example.
First, to find critical numbers, we need to compute the derivative of the function. The derivative, denoted f'(x), shows the rate at which the function is changing:
Step 1: Derivative Calculation
For our function, the derivative is:
f'(x) = 15x^2 – 2x
Step 2: Setting the Derivative to Zero
Next, we set the derivative equal to zero to find critical numbers:
15x^2 – 2x = 0
This can be factored:
x(15x – 2) = 0
From this equation, we can find:
- x = 0
- 15x – 2 = 0 ⟹ x = 2/15
In this case, the function does have critical numbers. If a function ultimately has no critical numbers, it could mean several things:
- Constant Function: The function could be a constant, meaning it doesn’t change regardless of the input values. For instance, f(x) = 5 has no critical points since its derivative is zero everywhere.
- No Turning Points: The function may be a monotonically increasing or decreasing function over its entire domain without any local extrema. For example, f(x) = e^x is always increasing and never zeroes its derivative.
- Undefined Derivative: It could also mean that the derivative is undefined at certain points, but those points are not within the domain or do not fall into the category of critical points due to being endpoints of a closed interval.
In conclusion, when analyzing a function like f(x) = 5x^3 – x^2 + 25 that has critical numbers, we find the points where its derivative equals zero. However, if we considered a different function without critical numbers, we would find that it could either be constant, continually increasing/decreasing, or undefined in a way that does not lead to critical points in practical terms. This concept plays a crucial role in calculus as it helps us understand the behavior of functions in different scenarios.