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I’m struggling with questions c, e and f.
I don’t think I understand how to find stationary points.
I don’t think I understand how to find stationary points.
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skeeter said:$L(\lambda) = \lambda^{150}e^{-3\lambda}$
$L’(\lambda) = 150 \lambda^{149} e^{-3\lambda} - 3\lambda^{150} e^{-3\lambda}$
$L’(\lambda) = 3\lambda^{149}e^{-3\lambda} (50-\lambda)$
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 $\lambda = 50$
$\lambda <50 \implies L’ > 0 \implies L$ 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
Country Boy said:$L’(\lambda) = 0$ at $\lambda = 50$ OR at $\lambda= 0$
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.
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.
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.
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.
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.