Understanding Relative Minimums | Calculus Example and Explanation

With respect to the derivative of f(x), when do relative minimums occur?

In calculus, relative minimums occur in a function f(x) at points where the derivative of f(x) changes from negative to positive

In calculus, relative minimums occur in a function f(x) at points where the derivative of f(x) changes from negative to positive. This means that the slope of the function changes from decreasing to increasing. Formally, a point x = c is considered a relative minimum if f'(c) = 0 and f”(c) > 0, where f'(x) represents the derivative of f(x), and f”(x) represents the second derivative.

To understand this concept better, let’s break it down with an example:

Consider the function f(x) = x^3 – 6x^2 + 9x + 2. To find the relative minimums, we need to take the derivative of f(x) and find the points where the derivative changes from negative to positive.

First, find the derivative of f(x):
f'(x) = 3x^2 – 12x + 9

Next, set f'(x) equal to 0 to find the critical points:
3x^2 – 12x + 9 = 0

Factoring the quadratic equation gives us:
(3x – 3)(x – 3) = 0

Setting each factor equal to 0:
3x – 3 = 0 => x = 1
x – 3 = 0 => x = 3

So, we have two critical points: x = 1 and x = 3.

Now, evaluate the second derivative of f(x):
f”(x) = 6x – 12

Evaluate f”(x) at each critical point:
f”(1) = 6(1) – 12 = -6
f”(3) = 6(3) – 12 = 6

Here, we notice that f”(1) is negative, and f”(3) is positive.

Therefore, at x = 1, the slope changes from decreasing to increasing, indicating a relative minimum. Similarly, at x = 3, the slope changes from increasing to decreasing indicating another relative minimum.

To summarize, the relative minimums of f(x) occur at x = 1 and x = 3, where the derivative changes from negative to positive (f”(x) > 0).

More Answers: