Find the range of values of the constant a at which the equation x^3-3a^2x+2=0 has 3 different real number roots.

Luisottifp 2022-09-25 Answered
Find range of values
Find the range of values of the constant a at which the equation x 3 3 a 2 x + 2 = 0 has 3 different real number roots.
I took the derivative and found that x = a , a. Then I solved for f ( a ) = 0 and f ( a ) = 0 to find that a = 1 , 1.
How do I use this information to find the range of values, or am I on the wrong path completely?
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Answers (2)

Simon Zhang
Answered 2022-09-26 Author has 7 answers
Step 1
You're on the wrong path. The points where f = 0 are "flat spots" in the graph. That doesn't tell you anything about where it's zero.
Actually, it tells you a little: between any two zeroes, there has to be a flat spot. So if a = 0, you can't possibly have three zeroes, because you've got only one flat spot, at 0.
Step 2
On the other hand, you know that f ( a ) = 0 and f ( a ) = 0, and the graph's a cubic, i.e., it goes from lower left to upper right. If it happens that f ( a ) > 0 and f ( a ) < 0, then by the intermediate value theorem, f must have three zeroes.
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Elias Heath
Answered 2022-09-27 Author has 2 answers
Step 1
Since f ( x ) = 3 x 2 3 a 2 = 3 ( x a ) ( x + a ) you see that f is increasing on ( , | a | ] and on [ | a | , + ) and decreasing in [ | a | , | a | ]. On every such interval you can have at most one solution to the equation f ( x ) = 0, because of monotonicity. To have a solution in each interval you need the function to change sign on the extreme points of the interval.
Step 2
So to have three solutions you need f ( | a | ) > 0 and f ( | a | ) < 0 which gives:
2 | a | 3 + 2 > 0 2 | a | 3 + 2 < 0
which is true for | a | > 1.
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