Question

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!

Answer 1

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

Answer 2

Can't i figure out if its increasing or decreasing using the first derivative test?
What is the point of the second derivative test?