To determine whether the function [tex]\( f(x) = x^2 \)[/tex] increases or decreases when [tex]\( x > 1 \)[/tex], we can analyze its derivative.
1. Define the function:
[tex]\[ f(x) = x^2 \][/tex]
2. Compute the derivative of [tex]\( f(x) \)[/tex]:
Using the power rule for differentiation, which states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( nx^{n-1} \)[/tex], we find:
[tex]\[ f'(x) = \frac{d}{dx}(x^2) = 2x \][/tex]
3. Analyze the sign of the derivative for [tex]\( x > 1 \)[/tex]:
We need to check the value of [tex]\( f'(x) \)[/tex] when [tex]\( x > 1 \)[/tex]. Specifically:
[tex]\[ f'(x) = 2x \][/tex]
For [tex]\( x > 1 \)[/tex], let's consider a point slightly greater than 1, say [tex]\( x = 1.1 \)[/tex].
4. Substitute [tex]\( x = 1.1 \)[/tex] into the derivative:
[tex]\[ f'(1.1) = 2 \times 1.1 = 2.2 \][/tex]
Since [tex]\( f'(1.1) = 2.2 \)[/tex] is positive, we can conclude that [tex]\( f(x) = x^2 \)[/tex] is increasing when [tex]\( x > 1 \)[/tex]. This is because the positive derivative indicates that the function's slope is positive, implying that the function is increasing in this interval.
