Question

where is the function ( f(x)=x^{2} ) increasing? where is it decreasing?
the function ( f(x)=x^{2} ) is increasing on the interval ( (0, infty) ).
(type your answer in interval notation.)
the function ( f(x)=x^{2} ) is decreasing on the interval
(type your answer in interval notation.)

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=x^{2}\) is \(f^{\prime}(x) = 2x\) (using the power rule \((x^{n})^\prime=nx^{n - 1}\)).

Step2: Determine where the derivative is positive (increasing) and negative (decreasing)

Set \(f^{\prime}(x)>0\):
\(2x>0\) implies \(x > 0\).
Set \(f^{\prime}(x)<0\):
\(2x<0\) implies \(x<0\).

Answer:

The function \(f(x)=x^{2}\) is decreasing on the interval \((-\infty,0)\)