I have the equation:
cos x - x = f(x)
I am told to find the relative extrema. I am also told to use the Second Derivative Test where applicable.
My question is how do I solve this problem, and how do I know when to use the Second Deriv test?
Thanks!
Points where the first derivative is zero are relative maxima if the second derivatve is negative and relative minima if the second derivative is positive.
In your case, df/dx = 0 when f'(x) = -sin x -1 = 0; sin x = -1
That happens when x = 3 pi/2.
At that point, f''(x) = -cos x = 0
So the second derivative fails to show if is a maximum or minimum. 3 pi/2 could be an inflection point, where the curve flattens out and then resumes its downward trend
Can't i figure out if its increasing or decreasing using the first derivative test?
What is the point of the second derivative test?
To find the relative extrema of the equation cos(x) - x = f(x), you'll need to follow these steps:
Step 1: Find the derivative of f(x) with respect to x.
The derivative of f(x) will give you the slope of the function at any given point.
Step 2: Set the derivative equal to zero and solve for x.
To find the critical points where the slope is zero, you need to set the derivative equal to zero and solve for x. These critical points may potentially be the locations of relative extrema.
Step 3: Evaluate the second derivative of f(x).
The second derivative of f(x) measures the concavity of the function at any given point. It helps us determine whether the critical points found in step 2 are relative maxima, relative minima, or inflection points.
Step 4: Apply the Second Derivative Test (where applicable).
You can use the Second Derivative Test to analyze the nature of the critical points. If the second derivative is positive at a particular point, then the function has a relative minimum at that point. If the second derivative is negative, then there is a relative maximum at that point. If the second derivative is zero, the test is inconclusive, and further analysis may be required.
In this specific case, you need to find the second derivative of f(x) to understand whether the critical points are relative maxima, relative minima, or if the test is inconclusive.
To determine when to use the Second Derivative Test, check whether the second derivative is continuous and defined for the critical points. If it is defined, you can use the test. If it is not defined, you may need to use an alternative method to analyze the nature of the critical points.
Remember, the Second Derivative Test is not always conclusive, and additional analysis may be required to obtain complete information about extrema.