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Determine if each of the following conjectures is always true, sometimes true, or never true. Be sure to provide examples and/or counterexamples to support your claim.

Conjecture 1: If the slope of the first derivative of a function is negative over the interval then a local maximum of that function will exist somewhere in that interval.

The slope of the first derivative is really the second derivative.
If the second derivative is negative, the graph of the function is concave down.
Does this guarantee that there is a local maximum?

Conjecture 2: Local extrema of a function are found where the derivative equals zero.

Compare the derivative values at the local extrema of and .

Conjecture 3: Local minima exist at the -values where the first derivative changes from positive to negative.

Will this statement be true if the function has points of discontinuity? Will this be true if the function has points of non-differentiability?

Conjecture 4: When the first derivative and the second derivative both equal zero, then the function has both a local extrema and an inflection point.

If the derivatives exist at a point, then the function must be smooth and continuous at the point. Can a smooth, continuous curve have a point that is a maxima/minima and a point of inflection simultaneously?