Science:Math Exam Resources/Courses/MATH102/December 2011/Question 02 (c)/Solution 1

To draw a qualitatively accurate sketch of the derivative of the function (i.e: $f'(x)$), we need to observe some qualitative features of the slope:

(1) We notice that the slope of the function is zero (i.e: function has a horizontal tangent line) at $x=-1$ and $x=1$. (i.e: $f'(x)=0$ at $x=-1$ and $x=1$)

(2) We notice the function has positive slope on $(-1,1)$ (i.e: $f'(x)>0$ on $(-1,1)$)

(3) We notice the function has negative slope on $(-3,-1)\cup (1,3)$ (i.e: $f'(x)<0$ on $(-3,-1)\cup (1,3)$).

So we need to sketch a curve for $f'(x)$ that the properties described in points (1)-(3). Such a curve is plotted in black below.