To determine the interval of increase for the quadratic function [tex]\(f(x) = x^2\)[/tex], let's analyze its behavior step by step.
1. Understand the nature of the quadratic function [tex]\( f(x) = x^2 \)[/tex]:
- Quadratic functions are parabolic in shape.
- The general form of a quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex]. In this specific case, [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 0 \)[/tex], so the function simplifies to [tex]\( f(x) = x^2 \)[/tex].
2. Find the derivative of [tex]\( f(x) = x^2 \)[/tex] to determine its rate of change:
- The derivative [tex]\( f'(x) \)[/tex] gives the slope of the tangent line to the function at any point [tex]\( x \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex], the derivative is:
[tex]\[
f'(x) = \frac{d}{dx}(x^2) = 2x
\][/tex]
3. Analyze the sign of the derivative [tex]\( f'(x) = 2x \)[/tex]:
- The sign of [tex]\( f'(x) \)[/tex] indicates whether the function [tex]\( f(x) \)[/tex] is increasing or decreasing.
- If [tex]\( f'(x) > 0 \)[/tex], [tex]\( f(x) \)[/tex] is increasing.
- If [tex]\( f'(x) < 0 \)[/tex], [tex]\( f(x) \)[/tex] is decreasing.
4. Determine when [tex]\( f'(x) = 2x \)[/tex] is positive:
- Set [tex]\( 2x > 0 \)[/tex]:
[tex]\[
x > 0
\][/tex]
- So, the derivative [tex]\( f'(x) \)[/tex] is positive when [tex]\( x > 0 \)[/tex].
5. Conclusion:
- The function [tex]\( f(x) = x^2 \)[/tex] is increasing on the interval where [tex]\( x > 0 \)[/tex].
Therefore, the interval of increase for the function [tex]\( f(x) = x^2 \)[/tex] is:
[tex]\[
\{x \mid x > 0\}
\][/tex]
In other words, the function [tex]\( f(x) = x^2 \)[/tex] increases for values of [tex]\( x \)[/tex] that are greater than 0.
