Differentiation help (stationary points)

In summary, we discussed the concept of stationary points, which occur where the derivative is zero or undefined. For the function $L(\lambda) = \lambda^{150}e^{-3\lambda}$, its derivative $L'(\lambda)$ is defined everywhere in its given domain. By using the 1st derivative test for extrema, we can find the maximum point at $\lambda = 50$ by evaluating $L'(\lambda)$ and determining that it is equal to 0. Additionally, we can use the 2nd derivative test for extrema by evaluating $L''(\lambda)$ to determine if the stationary point is a maximum or minimum. Finally, we also discussed considering the log function for finding stationary points,
  • #1
Lhh
3
0
I’m struggling with questions c, e and f.
I don’t think I understand how to find stationary points.

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  • #2
stationary points occur where the derivative is zero or is undefined.
in this case, $L’$ is defined everywhere in the function’s given domain.

skeeter said:
$L(\lambda) = \lambda^{150}e^{-3\lambda}$

$L’(\lambda) = 150 \lambda^{149} e^{-3\lambda} - 3\lambda^{150} e^{-3\lambda}$

$L’(\lambda) = 3\lambda^{149}e^{-3\lambda} (50-\lambda)$

$L’(\lambda) = 0$ at $\lambda = 50$

$\lambda <50 \implies L’ > 0 \implies L$ is increasing.
$\lambda >50 \implies L’ <0 \implies L$ is decreasing
therefore, $L$ has a maximum at $\lambda =50$
this method is called the 1st derivative test for extrema.

You could also evaluate the value of $L’’(50)$ to determine if the stationary point is a maximum or minimum.
if $L’’(50) <0$, then $L(50)$ is a maximum
if $L’’(50) >0$, then $L(50)$ is a minimum
this is the 2nd derivative test for extrema.

see what you can get done with the log function
 
  • #3
skeeter said:
stationary points occur where the derivative is zero or is undefined.
in this case, $L’$ is defined everywhere in the function’s given domain.
$L’(\lambda) = 0$ at $\lambda = 50$

$L’(\lambda) = 0$ at $\lambda = 50$ OR at $\lambda= 0$

$\lambda <50 \implies L’ > 0 \implies L$ is increasing.
$\lambda >50 \implies L’ <0 \implies L$ is decreasing
therefore, $L$ has a maximum at $\lambda =50$
this method is called the 1st derivative test for extrema.

You could also evaluate the value of $L’’(50)$ to determine if the stationary point is a maximum or minimum.
if $L’’(50) <0$, then $L(50)$ is a maximum
if $L’’(50) >0$, then $L(50)$ is a minimum
this is the 2nd derivative test for extrema.

see what you can get done with the log function
 
  • #4
Country Boy said:
$L’(\lambda) = 0$ at $\lambda = 50$ OR at $\lambda= 0$

$\lambda = 0$ is not in the given domain of the function ... reference the problem statement in post #1
 

FAQ: Differentiation help (stationary points)

1. What is a stationary point in differentiation?

A stationary point in differentiation is a point on a curve where the gradient is zero. This means that the tangent line at that point is horizontal and the curve is neither increasing nor decreasing at that point. Stationary points can be either maximum points, where the curve changes from increasing to decreasing, or minimum points, where the curve changes from decreasing to increasing.

2. How do you find stationary points in differentiation?

To find stationary points in differentiation, you must first take the derivative of the function. Then, set the derivative equal to zero and solve for the variable. The values of the variable that satisfy this equation will be the x-coordinates of the stationary points. To determine if the stationary points are maximum or minimum points, you can use the second derivative test or check the concavity of the curve at those points.

3. What is the significance of stationary points in differentiation?

Stationary points are important in differentiation because they can help us identify maximum and minimum points on a curve. These points are often used in optimization problems to find the maximum or minimum value of a function. They also allow us to analyze the behavior of a function and determine if it is increasing or decreasing at a certain point.

4. Can a function have more than one stationary point?

Yes, a function can have more than one stationary point. In fact, a function can have multiple stationary points at different locations on the curve. It is also possible for a function to have no stationary points, if the derivative is never equal to zero.

5. How do you determine if a stationary point is a maximum or minimum point?

To determine if a stationary point is a maximum or minimum point, you can use the second derivative test. If the second derivative is positive, the stationary point is a minimum point. If the second derivative is negative, the stationary point is a maximum point. If the second derivative is zero, the test is inconclusive and you may need to use other methods, such as checking the concavity of the curve, to determine the nature of the stationary point.

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