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
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)
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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K has many values