<p>The following five point lie on a function: (1,20), (2,4), (5,3),(6,2), and (10,1). Find an equation that passes through these points and has these features:
a. There are at least three inflection points.
b. There is at least one local maximum.
c. There is at least one local minimum.
d. At least one critical point is not at a given point.
e. The curve is continuous and differentiable throughout.
f. The equation is not a single polynomial, but must be a piece-wise defined function.</p>
<p>I’m not going to do your homework for you (nor is anyone here), but here’s some tips to start you out:</p>
<p>"3 Inflections Points: Recall that an inflection point is where you have a change in concavity, usually found by setting the 2nd derivative equal to zero and solving. If you need 3 solutions, your 2nd derivative should have x^3 in it…which means your original function is probably an x^5.</p>
<p>1 Local Maximum/Minimum: If you have 3 Inflection Points, maximums will occur naturally, recall it’s where you have a change in increasing/decreasing…or where it’s concave down. You’ll probably have 3 - 6 maximums naturally, so don’t worry too much about this one.</p>
<p>1 Critical Point Not Given: This will also occur naturally if you have 3 inflection points.</p>
<p>Continuous and Differentiable: Make sure you don’t have any fractions or square roots, otherwise you’ll have discontinuities (Vertical Asymptotes, Domain Problems). This is a complicated way of saying, make sure you have a polynomial for an answer.</p>
<p>Piecewise: I said polynomial…now they throw a wrench in and say make it a piecewise. So just define your domain, break the polynomial into pieces and make a piecewise based on your domain."</p>
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