IB Maths AA SL 1.6 Deductive Proof Questions

Practise writing direct algebraic proofs of identities, divisibility and parity by showing each justified rearrangement, assumption and exceptional case.

Exam points

Rearrange and simplify numerical or algebraic expressions line by line to prove a stated identity or equality, using equality or identity notation correctly.
Represent integers such as consecutive values algebraically and use expansion, factorisation or modular reasoning to prove divisibility or parity claims.
Write a complete deductive proof from the stated assumptions, showing each justified step and checking the resulting formula or exceptional case.

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[Maximum number: 2]

The derivative of a function f is given by $f^{\prime}(x)=\frac{2 x+2}{x^{2}+2 x+2}$ , for $x \in \mathbb{R}$ .

Show that $x^{2}+2 x+2>0$ for all values of x.

METHOD 1
$x^{2}+2 x+2=(x+1)^{2}+1$ $(x+1)^{2} \geq 0$ $(x+1)^{2}+1>0$

METHOD 2
discriminant $\Delta=4-8(=-4)$ $\Delta<0$ and concave up OR $\Delta<0$ and coefficient of $x^{2}>0$ (may be seen in diagram)
hence $x^{2}+2 x+2>0$

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IB Maths AA SL 1.6 Deductive Proof Questions question 1