Ian B. answered • 04/21/19
Yale Senior for Academic and Test Prep Help
The minima and maxima of a function happen at the critical points, or when the derivative of the function is equal to zero. This makes sense, because a local minima or maxima will happen at a turnaround, where the slope of the graph is equal to zero.
Let's start by saying that F'(x)=3x^2 - 12 x + 9 given the power rule.
We can now set F'(x)=0. This factors into 3(x-3)(x-1), which allows us to see that the critical points happen at x=1, x=3.
Now we must determine whether each point is a maximum, minimum or neither. I do this by placing 1 and 3 on a number line. I pick a number between them (let's say 2) and evaluate F'(2)= -3. Because this is negative, I know that every x-value between 1 and 3 will have a negative y-value. I can confirm that numbers less than 1 and greater than 3 will be positive by picking any number in that range and plugging it in to F'(x).
Now, what this tells us is that at x=3, the derivative goes from positive to negative. This means the slope of the original function goes from increasing to decreasing, which translates to a maximum. At x=3, the derivative goes from negative to positive, which means the slope of the original function goes from decreasing to increasing. This translates to a minimum.
To summarize, the critical points are x=1, which will be a maximum, and x=3, which will be a minimum. Please let me know if you have any questions, or would like to work together on future calculus questions!
