Find the coordinates of the turning point and determine wether it is minimum or maximum.
y=xlnx-2x
The answer in the book says the co-ordinates are (e,-e), the closest I have come is (1/lnx,-1/lnx) which works if 1/lnx=e, but I don't think that it does. This question is in relation to derivatives.
To find the coordinates of the turning point and determine whether it is a minimum or maximum, we need to find the critical point of the function first.
The given function is y = xln(x) - 2x.
To find the critical point, we need to find where the derivative of the function is equal to zero or undefined. Let's first find the derivative of the function.
Taking the derivative of y with respect to x using the product rule, we get:
dy/dx = d/dx (x) * ln(x) + x * d/dx (ln(x)) - 2
Simplifying this expression gives:
dy/dx = ln(x) + x * (1/x) - 2
dy/dx = ln(x) + 1 - 2
dy/dx = ln(x) - 1
Now, we need to find where dy/dx is equal to zero or undefined. Setting dy/dx to zero and solving for x gives:
ln(x) - 1 = 0
ln(x) = 1
x = e
Therefore, x = e is a critical point of the function.
To determine whether it is a minimum or maximum, we need to analyze the concavity of the function. To do this, we can find the second derivative of the function.
Taking the derivative of dy/dx with respect to x, we get:
d²y/dx² = d/dx (ln(x) - 1)
Simplifying this expression gives:
d²y/dx² = 1/x
Now, we need to evaluate the second derivative at the critical point x = e.
d²y/dx² = 1/e
Since the second derivative is positive at x = e (1/e is positive), the turning point is a minimum.
Therefore, the coordinates of the turning point and whether it is a minimum or maximum are:
Coordinates: (e, -e)
Type: Minimum