IB Maths AA SL 1.6 Deductive proof Question Bank

Practise IB Maths AA SL 1.6 by constructing direct algebraic proofs of identities, divisibility and integer properties.

Exam points

Rearrange algebraic expressions line by line to prove a stated identity or equality.
Use consecutive integers or modular reasoning to show that an expression is divisible by a given integer.
Write a complete proof with clearly stated assumptions and no unsupported algebraic step.

[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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SL 1.6—Deductive proof question 1