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Mathematical Expression Editor
In this section we learn to find the critical numbers of a function.
1 Critical Numbers
In this section we will find the critical numbers of a given function. Critical
numbers come in two types and the idea behind the definition comes from the
possibilities at a local extreme.
The function has a critical number at if is defined and either
Type 1 critical numbers correspond to horizontal tangent lines in the graph of the
function. Type 2 critical numbers typically correspond to corner points or vertical
tangent lines.
2 Finding Critical Numbers
example 1 Find the critical numbers of the function Solution: We need to compute .
We have Noting that is defined for all values of , there are no type 2 critical
numbers. To find the type 1 critical numbers, we solve the equation Geometrically,
these are the points where the graph of has horizontal tangent lines. We get
Hence has two critical numbers: and . They are both of type 1, correspoding to
horizontal tangent lines.
(problem 1a) Find the critical numbers of the function .
Solve for
Are there any points where is undefined? If so, is defined at these points?
The critical numbers of are
no critical numbers
(problem 1b) Find the critical numbers of the function
Solve for
Are there any points where is undefined? If so, is defined at these points?
The critical numbers of are
and and no critical numbers
example 2 Find the critical numbers of the function Solution: We need to compute . We have Noting that is defined for all values of ,
there are no type 2 critical numbers. To find the type 1 critical numbers, we solve the
equation Geometrically, these are the points where the graph of has horizontal
tangent lines. We get Hence has two critical numbers, and , and they are both
type 1.
(problem 2) Find the critical numbers of the function
The critical numbers of are
and no critical numbers
example 3 Find the critical numbers of the function We need to compute using the
product and chain rules. We have Noting that is defined for all values of , there are
no type 2 critical numbers. To find the type 1 critical numbers, we solve the equation
Geometrically, these are the points where the graph of has horizontal tangent lines.
We get Note that the equation has no solutions since an exponential function is
always positive.
Hence has two critical numbers, and , and they are both type 1.
(problem 3a) Find the critical numbers of the function
Compute using the Product Rule
The critical numbers of are
and no critical numbers
(problem 3b) Find the critical numbers of the function
Compute using the Product Rule
The critical numbers of are
and no critical numbers
example 4 Find the critical numbers of the function We need to compute using the
quotient rule. We have Noting that is defined for all values of (since the
denominator is never equal to 0), there are no type 2 critical numbers. To find
the type 1 critical numbers, we solve the equation Geometrically, these
are the points where the graph of has horizontal tangent lines. We get
Hence has two critical numbers, and , and they are both type 1.
(problem 4a) Find the critical numbers of the function
Compute using the Quotient Rule
The critical numbers of are
and no critical numbers
(problem 4b) Find the critical numbers of the function
Compute using the Quotient Rule
The critical numbers of are
and no critical numbers
example 5 Find the critical numbers of the function We need to compute . We have
In this case, is undefined (division by zero). Hence is a critical number if is defined.
We can easily check this: , so it is defined and now we can conclude that is a type 2
critical number. To find the type 1 critical numbers, we solve the equation
Geometrically, these are the points where the graph of has horizontal tangent
lines. We get So there are no solutions. The function has no type 1 critical
numbers.
Hence has only one critical number, 0, and it type 2, where the function is not
differentiable. Geometrically, the function has a vertical tangent line at the critical
number .
(problem 5a) Find the critical numbers of the function
The derivative can be written as
The critical numbers of are
no critical numbers
(problem 5b) Find the critical numbers of the function
Rewrite as a power function by adding exponents
The critical numbers of are
no critical numbers
example 6 Find the critical numbers of the function We need to compute
. We have In this case, is undefined at (division by zero). Hence, if is
defined then would be a type 2 critical number. However, we can easily see
that , is undefined, so that is a not in the domain of and hence it is not a
critical number. To find the type 1 critical numbers, we solve the equation
Geometrically, these are the points where the graph of has horizontal tangent
lines. We get So there are no solutions. The function has no type 1 critical
numbers.
Hence has no critical numbers.
(problem 6a) Find the critical numbers of the function
The critical numbers of are
no critical numbers
(problem 6b) Find the critical numbers of the function in the interval
The critical numbers of are
and no critical numbers
Here is a detailed, lecture style video on critical numbers: