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Derivative proofs

Please state whether each statement is true or false, and if false please explain why

The maximum slope of the graph of y = sin(bx) is b

If f''(2) = 0, then the graph of f must have a point of inflection at x = 2.

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Dear student,

1. y' = b cos(bx)
y" = -b^2 sin(bx)
Maxima or minima of y' occur when y" = 0, so 0 = -b^2 sin(bx)
so sin(bx) = 0 for a maximum slope of y, so bx = 0 or bx = pi.
If bx = 0 then x = 0 (assuming b<>0, which must be true or y = 0!). If x=0 then ...

Solution Summary

This shows how to determine the truth of two statements regarding functions and differentiations.