Question

below is the function ( f(x) ).
over which interval of ( x ) values is ( f^{prime}>0 )?
( (1, infty) )
( 1, infty) )
( (-infty, 1) )
( (-infty, 1 )
( (-infty, infty )
over which interval of ( x ) values is ( f^{prime}<0 )?
( (1, infty) )
( 1, infty) )
( (-infty, 1) )
( (-infty, 1 )
( (-infty, infty )

Explanation:

Step1: Recall the relationship between the derivative and the function's slope

The derivative \(f^{\prime}(x)\) represents the slope of the tangent line to the function \(y = f(x)\). If \(f^{\prime}(x)>0\), the function is increasing (slope of the tangent line is positive). If \(f^{\prime}(x)<0\), the function is decreasing (slope of the tangent line is negative).

Step2: Analyze the graph for the increasing - interval

Looking at the graph of the parabola \(y = f(x)\), we can see that for \(x>1\), as \(x\) increases, \(y = f(x)\) also increases. So, \(f^{\prime}(x)>0\) when \(x\in(1,\infty)\).

Step3: Analyze the graph for the decreasing - interval

For \(x < 1\), as \(x\) increases, \(y=f(x)\) decreases. So, \(f^{\prime}(x)<0\) when \(x\in(-\infty,1)\).

Answer:

For \(f^{\prime}(x)>0\): \((1,\infty)\)
For \(f^{\prime}(x)<0\): \((-\infty,1)\)