Continuing with our review in mathematics, we cover another AP subject that multiple students take classes for, AP Calculus AB. This course is one of the most frequently taken courses in any college or university (usually taught as Calculus 1 at the college level). Hence the large enrollment of exam takers. A typical question that students will cover in the AP Calculus AB exam will be working with derivatives and specifically how to interpret results of derivatives. Let's take a look at the example below that deals with this:
Which of the statements below about the function f(x)=2x4−4x2 is true?
Your answer choices are:
- The function has no inflection points
- Its graph has one points of inflection and three local extrema
- Its graph has two points of inflection and two local extrema
- Its graph has one points of inflection and one local extrema
- Its graph has three points of inflection and one local extremum
The answer to this question is Its graph has two points of inflection and two local extrema. To solve this question you need to understand what points of inflection and local extrema are. Local extremas are also known as local minimums and maximums of a graph, which are the low and high points of a curve where the slope at that point is equal to zero. One can easily solve for this by taking the first derivative of the function and setting it equal to zero. You can then find which values of x will result in a slope of zero.
The derivative of the function is f '(x)=8x3−8x. Setting that equal to zero and solving for x gives us local extremas at x= +-1, indicating that there are two local extrema for this function.
To solve for points of inflection, one must take the second derivative of a function, set it equal to zero and solve for x. The 2nd derivative is f ''(x)=24x2−8. Setting this equal to zero and solving for x gives us two solutions of x=+1/√3. Therefore, there are two points of inflection.
Lookinng at our answer choices again, there is only one choice that is true and we just proved the answer. Success!
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