Example 3:

Find the maximum and minimum of the following function and then sketch it. \[f(x) = 3x^3 +3x^2\]

Answer:

First differentiate the function.
\[f'(x) =9x^2 +6x\] We know that the derivative of the function is zero at any maximum or minimum. Let \(f'(x)=0\) to find where these points are. \[f'(x)=9x^2 +6x=0\]\[3x(3x+2)=0\] \[\implies x=0 \text{ or } x=-\frac{2}{3} \]

Now we must decide whether these two points are maxima or minima or points of inflection.
This is done by finding the second derivative. \[f''(x) = 18x + 6\] Subbing in the two values for x calculated above we get \[f''(0) = 18(0) + 6=6\]\[f''(-\frac{2}{3}) = 18(-\frac{2}{3})+6=-12+6=-6\]
So this means that \(f''(x)>0 \) for \(x=0\) and \(f''(x)<0\) for \( x=-\frac{2}{3} \). Therefore, we now know that \( x=-\frac{2}{3} \) is a local maximum and \(x=0\) is a local minimum.

Finally, before we graph the function, it is necessary to factorise it and let it equal to zero in order to determine the points where it crosses the axes. \[f(x) = 3x^3 + 3x^2 = 3x^2(x+1)=0\] \[\implies x=0 \text{ or } x=-1\].

The resulting graph can be shown below.