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Questions › DSE › DSE 2010 › Q34
2 marksMathematics › Limits, continuity and differentiability › Intermediate value theorem, monotonicity and rootsHardabout 4 min

DSE 2010 Question 34: Limits, continuity and differentiability

Suppose f:ℜ2→ℜf:\Re^{2}\to\Ref:ℜ2→ℜ is increasing in both arguments. For x,y∈ℜx,y\in\Rex,y∈ℜ let x∧y=xx\wedge y=xx∧y=x if x≤yx\le yx≤y and =y=y=y if x>yx>yx>y. Define g:ℜ2→ℜg:\Re^{2}\to\Reg:ℜ2→ℜ by g(x,y)=f(x,y)−12f(x∧y,x∧y)g(x,y)=f(x,y)-\tfrac{1}{2}f(x\wedge y,x\wedge y)g(x,y)=f(x,y)−21​f(x∧y,x∧y) if x≥yx\ge yx≥y, and g(x,y)=12f(x∧y,x∧y)g(x,y)=\tfrac{1}{2}f(x\wedge y,x\wedge y)g(x,y)=21​f(x∧y,x∧y) if x<yx<yx<y. Which statement is correct?

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Plain-text version

Suppose f:ℜ²→ℜ is increasing in both arguments. For x,y∈ℜ let x∧ y=x if x≤ y and =y if x>y. Define g:ℜ²→ℜ by g(x,y)=f(x,y)-1/2f(x∧ y,x∧ y) if x≥ y, and g(x,y)=1/2f(x∧ y,x∧ y) if x<y. Which statement is correct? (A) g is increasing in x and decreasing in y (B) g is increasing in both x and y (C) g is increasing in x but may or may not be increasing in y (D) g may or may not be increasing in x

Also written as: Suppose f: Re^2-> Re is increasing in both arguments. For x,y in Re let x and y=x if x<= y and =y if x>y. Define g: Re^2-> Re by g(x,y)=f(x,y)-1/2f(x and y,x and y) if x>= y, and g(x,y)=1/2f(x and y,x and y) if x<y. Which statement is correct?

Limits, continuity and differentiability has come up 122 times in 7 exams since 2006.

What this question tests

This question is on Intermediate value theorem, monotonicity and roots, part of Limits, continuity and differentiability in Mathematics.

Using continuity and the sign of the derivative to show roots exist or are unique.

TestStop has 12 past-paper questions on this subtopic.

Practise Limits, continuity and differentiabilityMore on Intermediate value theorem, monotonicity and roots

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