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

Construct and evaluate deductive proofs using definitions, algebraic identities and logical implications.
Present a valid sequence of mathematical statements and justify conclusions using complete evidence.

[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