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Second Derivatives and Beyond

Second Derivatives and Beyond

At a Glance - Critical Points

We say x = c is a critical point of the function f if f (c) exists and '(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interesting. When climbing Mount Everest, we might say that we've reached the critical point when we've reached the summit. It's where we can enjoy the view while feasting on granola.

Be Careful: A critical point of a function f is a value in the domain of f at which the derivative is 0 or undefined.

It's also possible for a function to have no critical points at all.

To find the critical points of a function f we

  • take the derivative of f
      
  • find any places where ' is undefined but f is defined, and finally
      
  • find the roots of the derivative.

We can often check our answers by graphing the function and making sure it looks like it has critical points in the right places. We know what a graph looks like at a spot where the derivative doesn't exist:

and we know what a graph looks like when it has a derivative of 0 (horizontal tangent line):

Example 1

Find the critical points of the function f (x) = 5x2 + 4x - 2.


Example 2

Find the critical points of the function

f (x) = 3x2/3 + 3x5/3


Example 3

Find the critical points of the function f (x) = ex.


Example 4

Find the critical points of the function f (x) = 1/x.


Exercise 1

For the function, find all critical points or determine that no such points exist.


Exercise 2

For the function, find all critical points or determine that no such points exist.

f (x) = cos x


Exercise 3

For the function, find all critical points or determine that no such points exist.


Exercise 4

For the function, find all critical points or determine that no such points exist.


Exercise 5

For the function, find all critical points or determine that no such points exist.

f (x) = xex


Exercise 6

For the function, find all critical points or determine that no such points exist.

f (x) = 4x3 + 3x2 – 2x + 1


Exercise 7

For the function, find all critical points or determine that no such points exist.

f (x) = sin x cos x


Exercise 8

For the function, find all critical points or determine that no such points exist.

f (x) = sin2 x


Exercise 9

For the function, find all critical points or determine that no such points exist.

f (x) = ex + e-x


Exercise 10

For the function, find all critical points or determine that no such points exist.

f (x) = 3x – 2x


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