121616949-math.113 - 5.3 The second derivative test 99 15...

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5.3 The second derivative test 99 15. Let f ( θ ) = cos 2 ( θ ) - 2 sin( θ ). Find the intervals where f is increasing and the intervals where f is decreasing inside of [0 , 2 π ]. Use this information to classify the critical points of f as either local maximums, local minimums, or neither. 16. Let r > 0. Find the local maxima and minima of the function f ( x ) = r 2 - x 2 on its domain ( - r, r ). Sketch the curve and explain why the result is unsurprising. 17. Let f ( x ) = ax 2 + bx + c with a = 0. Show that f has exactly one critical point using the first derivative test. Give conditions on a and b which guarantee that the critical point will be a maximum. It is possible to see this without using calculus at all; explain. The basis of the first derivative test is that if the derivative changes from positive to negative at a point at which the derivative is zero then there is a local maximum at the point, and similarly for a local minimum. If f changes from positive to negative it is decreasing; this means that the derivative of f , f
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