Question

part a. approximate when the function is positive, negative, increasing, or decreasing.
the function is
positive

Explanation:

Step1: Determine when the function is positive

A function \(y = f(x)\) is positive when \(y>0\). Looking at the graph, the function \(y = 2x^{3}-6x - 4\) is above the \(x -\)axis (where \(y = 0\)) when \(x>-2\) and \(x < - 1\).

Step2: Determine when the function is negative

A function \(y = f(x)\) is negative when \(y<0\). The function \(y=2x^{3}-6x - 4\) is below the \(x -\)axis (where \(y = 0\)) when \(-1

Step3: Determine when the function is increasing

A function \(y = f(x)\) is increasing when the slope of the tangent line (or the general trend of the graph) is positive. For the cubic function \(y = 2x^{3}-6x - 4\), we can also use the fact that for a cubic \(y=ax^{3}+bx^{2}+cx + d\) (\(a = 2>0\)), and by looking at the graph, the function is increasing when \(x<-1\) and \(x>1\).

Step4: Determine when the function is decreasing

A function \(y = f(x)\) is decreasing when the slope of the tangent line (or the general trend of the graph) is negative. Looking at the graph of \(y = 2x^{3}-6x - 4\), the function is decreasing when \(-1

Answer:

• Positive: \(x\in(-2,-1)\)
• Negative: \(x\in(-1,2)\)
• Increasing: \(x\in(-\infty,-1)\cup(1,\infty)\)
• Decreasing: \(x\in(-1,1)\)