8.2.1 Critical Points

Question 1

Critical Points

Answer 1

• Critical points are the places on a graph where the derivative equals zero or is undefined. Interesting things happen at critical points.
• To find critical points, take the derivative, set the derivative equal to zero and solve, and find values where the derivative is undefined.

Question 2

note

Answer 2

• Consider this graph of some complicated function. There seem to be several interesting points.
• At x 1 , x 2 , x 4 , and x 5 , the tangent lines are horizontal, so the derivative of the function is 0 at each of those values.
• At x 3 , the tangent line is vertical. Its slope is undefined, so the derivative of the function at that point is undefined, too.
• These points are called critical points. To identify them you need to solve for values that make the derivative equal to zero or undefined.
• Here is an example of a function that you might want to graph.
• To find its critical points, first take the derivative.
• Next, set the derivative equal to zero and solve.
• Finally, look for places where the derivative is undefined. In this case, the derivative is defined for all real numbers.
• You will need to use the power rule to take the derivative of this function.
• First, notice that there are no values of x that make the derivative equal to zero.
• Next, look for values of x for which the derivative is
  undefined. One way to do this is to set the denominator equal to zero and solve. In this case, the derivative is undefined for x = 0, even though the function itself is defined at that point. Therefore, you may have a cusp point at that x-value.

Question 3

Does this function have a critical point at x = 0?

Answer 3

Yes.

Question 4

Find all of the critical points of p (t).

p (t) = t ^2 + 5t + 6.

Answer 4

t=−5/2

Question 5

Find all of the critical points of the function f (x) = |3x|

Answer 5

x = 0