# STEP Paper 1 Q3 1998

#### CaptainBlack

##### Well-known member
Just to show that not everything in a STEP paper in difficult, this is an easy question:

Which of the following are true and which false? Justify your answers

(i) $$a^{\ln(b)}=b^{\ln(a)}$$, for all $$a,b \gt 0$$.

(ii) $$\cos(\sin(\theta))=\sin(\cos(\theta))$$, for all real $$\theta$$.

(iii) There exists a polynomial $$P$$ such that $$|P(\theta)-\cos(\theta)| \lt 10^{ -6 }$$ for all real $$\theta$$

(iv) $$x^4+3+x^{-4} \ge 5$$ for all $$x\gt 0$$.

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#### Amer

##### Active member
1) True take the ln for both sides

2) False take theta = 0

3) that true using Taylor expansion of cos(theta)

4) How to solve it ?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
3) that true using Taylor expansion of cos(theta)
Really?

4) How to solve it ?
I assume z should be replaced by x. One way is to express $x^4+x^{-4}$ through $x+x^{-1}$. One needs to know that $x+x^{-1}\ge2$ for x > 0.

#### chisigma

##### Well-known member
3) that true using Taylor expansion of cos(theta)
The question is that is true for all $\theta$... the function $\cos \theta$ is bounded in $\theta \in \mathbb{R}$, any polinomial $P(\theta)$ which is not a constant is unbounded in $\theta \in \mathbb{R}$...

Kind regards

$\chi$ $\sigma$

#### CaptainBlack

##### Well-known member
1) True take the ln for both sides
It is true, but that is not as it stands a valid explanation, you are assuming it true and deriving a truth, which is invalid logic. You need to start with a known truth and from that derive the equality you are seeking to justify.

2) False take theta = 0
Yes.

3) that true using Taylor expansion of cos(theta)
No, a Taylor expansion is not a polynomial, and a Taylor polynomial does not satisfy what is to be demonstrated for all $$\theta$$

4) How to solve it ?
$$f(x)=x^4+3+x^{-4}$$ is continuous and differentiable for $$x\gt 0$$, it goes to $$+\infty$$ at $$x=0$$ and as $$x\to \infty$$. It has one stationary point in $$(0,\infty)$$ at $$x=1$$, which therefore must be a minimum and $$f(1)=5$$

CB

#### Amer

##### Active member
$\ln(a)\ln(b) = \ln(b)\ln(a)$
$\ln(b^{\ln(a)}) = \ln(a^{\ln(b)})$

#### CaptainBlack

##### Well-known member
$\ln(a)\ln(b) = \ln(b)\ln(a)$
$\ln(b^{\ln(a)}) = \ln(a^{\ln(b)})$