Step 1: Understanding the Concept:
Monotonicity refers to the property of a function being either entirely non-increasing or non-decreasing.
In calculus, we use the first derivative to test for these properties.
A function is "strictly decreasing" on an interval if its slope (derivative) is strictly negative throughout that entire interval.
Graphically, this means that as you move from left to right along the x-axis, the y-values of the function are consistently getting smaller.
For a quadratic function (a parabola), there is typically one region where it decreases and one region where it increases, separated by the vertex (turning point).
Step 2: Key Formula or Approach:
1. Find the first derivative \( f'(x) \).
2. Set the inequality \( f'(x)<0 \) to satisfy the "strictly decreasing" condition.
3. Solve the resulting inequality for \( x \).
Step 3: Detailed Explanation:
Given the function:
\[ f(x) = x^2 - 4x + 6 \]
We differentiate the function using the power rule:
\[ f'(x) = 2x - 4 \]
The function is strictly decreasing in the region where the slope is less than zero:
\[ 2x - 4<0 \]
To solve this linear inequality, we first move the constant to the other side:
\[ 2x<4 \]
Next, divide both sides by 2:
\[ x<2 \]
This means that for any value of \( x \) smaller than 2, the function is moving downwards.
In standard mathematical interval notation, the set of all values less than 2 is written as:
\[ (-\infty, 2) \]
We use a round parenthesis at 2 because the question asks for "strictly" decreasing, meaning the point where the derivative is exactly zero (\( x = 2 \)) is excluded from the interval.
Step 4: Final Answer:
The interval of strict decrease is \( (-\infty, 2) \).