PRECALC Find K So That
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PRECALC Find K So That

[From: ] [author: ] [Date: 12-08-02] [Hit: ]
take the easiest case where k=0.which has a double root at x=0 and a single root at x=12.This is your two different zeroes case.Now sketch a graph to see what this looks like.From the left, it rises to a local maximum at x=0,......
X^3 -12X^2 + K

find K for
Two Different Zeros
Three Different Zeros ( I believe K=X for this one )
Only one Real Zero

Any help would be greatly appreciated thanks

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Answer: k = 0 gives two real roots, k = 1 gives three real roots, and k = -1 gives only one real root.

Here's one way to do it. First, take the easiest case where k=0. Then your function is

f(x) = x^3 - 12x^2 = x^2 (x - 12) = 0

which has a double root at x=0 and a single root at x=12. This is your "two different zeroes" case.

Now sketch a graph to see what this looks like. From the left, it rises to a local maximum at x=0, curves negative again, reaches a local minimum somewhere between 0 and 12, and then goes up to positive infinity, crossing the x-axis at x=12.

Now suppose we define a new function g(x) = f(x) - 1. When x=0, f(x)=0 and g(x) = f(x) - 1 = -1. This function g(x) never reaches the x-axis at x=0, but it still crosses the axis near x=12. When k = -1, then, you have exactly one real zero. (Answer)

Finally, if g(x) = f(x) + 1, the local max near x=0 is greater than zero. That means the function goes above the x-axis near x=0, reaches a local max g(x) > 0, then goes negative again before turning upward. When k=1, you have three distinct real roots. (Answer)

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1
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