Question 579322: sorry this is calculus, please help
Given the function f(x)=3cosx for the interval 0
i)determine the values of x for which the function is increasing
ii)find the critical points and determine their nature
iii)determine the concavity of the function, between the critical values
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  is continuous, but it does have maxima and minima.
To find where the function increases, decreases, or has a maximum or minimum, you need to look at its derivative.
To find curvature, points of inflection, concavity, you need to look at the second derivative.
The first derivative is
f'(x)=
It is zero at  ,  ,  , etc
Those would be maxima or minima of f(x).
The derivative is positive when  ,  ,  ,  , etc. Those are the intervals where f(x) is increasing. Where  ,  , the derivative is negative and the function decreases.
Putting it all together, the function increases from  to 0; has a maximum at x=0; decreases from there to  ; has a minimum at ; increases from there to  ; has a maximum at , and so on. The value of the function is  at all maxima and  at all minima.
The second derivative is f”(x)= .
It goes through zero and changes signs at  ,  ,  , etc. Those are points of inflection of f(x). The value of the first derivative at those points is either 3 or -3, which gives you the slope of the tangent at those points.
The second derivative is negative between  and  , positive between and  , negative between and  , and so on. As a result the function is concave downwards between and , concave upwards between and , concave downwards between and , and so on.
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