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Aleks04 [339]
3 years ago
5

If x^2+y^2=1, what is the largest possible value of |x|+|y|?

Mathematics
1 answer:
Marianna [84]3 years ago
6 0

If <em>x</em>² + <em>y</em>² = 1, then <em>y</em> = ±√(1 - <em>x</em>²).

Let <em>f(x)</em> = |<em>x</em>| + |±√(1 - <em>x</em>²)| = |<em>x</em>| + √(1 - <em>x</em>²).

If <em>x</em> < 0, we have |<em>x</em>| = -<em>x</em> ; otherwise, if <em>x</em> ≥ 0, then |<em>x</em>| = <em>x</em>.

• Case 1: suppose <em>x</em> < 0. Then

<em>f(x)</em> = -<em>x</em> + √(1 - <em>x</em>²)

<em>f'(x)</em> = -1 - <em>x</em>/√(1 - <em>x</em>²) = 0   →   <em>x</em> = -1/√2   →   <em>y</em> = ±1/√2

• Case 2: suppose <em>x</em> ≥ 0. Then

<em>f(x)</em> = <em>x</em> + √(1 - <em>x</em>²)

<em>f'(x)</em> = 1 - <em>x</em>/√(1 - <em>x</em>²) = 0   →   <em>x</em> = 1/√2   →   <em>y</em> = ±1/√2

In either case, |<em>x</em>| = |<em>y</em>| = 1/√2, so the maximum value of their sum is 2/√2 = √2.

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