HSC Stuff > HSC Mathematics Extension 2
Mathematics Extension 2 Challenge Marathon
Ali_Abbas:
--- Quote from: RuiAce on June 28, 2017, 05:13:03 pm ---A classic complex numbers question I just did a few minutes ago.
--- End quote ---
I believe I have a solution that differs from that of Rui's and is as follows.
RuiAce:
Tbh all I did was this.
RuiAce:
HintCan be made much easier using one of the tricks used in the previous problemSpoilerFind a contradiction!
Ali_Abbas:
--- Quote from: RuiAce on July 02, 2017, 03:35:52 pm ---Tbh all I did was this.
(Image removed from quote.)
--- End quote ---
You need to be careful when you apply the triangle inequality on a difference of terms as opposed to a sum of terms (occurring within the modulus signs). Although it will still give you a correct upper bound, it doesn't always give the true maximum of the expression. To see this, we can apply the triangle inequality on |alpha-1| + |alpha+1| giving:
But clearly 4 is not the true maximum so I'm not sure if your application of the triangle inequality qualifies as a concrete proof or if it merely gave the true maximum of the sum by shear coincidence.
RuiAce:
--- Quote from: Ali_Abbas on July 02, 2017, 08:23:21 pm ---You need to be careful when you apply the triangle inequality on a difference of terms as opposed to a sum of terms (occurring within the modulus signs). Although it will still give you a correct upper bound, it doesn't always give the true maximum of the expression. To see this, we can apply the triangle inequality on |alpha-1| + |alpha+1| giving:
But clearly 4 is not the true maximum so I'm not sure if your application of the triangle inequality qualifies as a concrete proof or if it merely gave the true maximum of the sum by shear coincidence.
--- End quote ---
The question did not require the least upper bound. I had explicitly stated that you need only prove this for 2sqrt(2).
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