Q3 from STEP 1998 Paper1

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

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

(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 \(x\gt 0\).

                                                                             

Solution:
(i) True, since we know for \(a,b \gt 0\) that \( \ln(b)\ln(a)=\ln(a)\ln(b) \) and by the laws of logarithms this may be rewritten: \(\ln(a^{\ln(b)})=\ln(b^{\ln(a)})\) and as the equality of logarithms implies the equality of the arguments we can conclude that \(a^{\ln(b)}=b^{\ln(a)}\) for all \(a,b \gt 0\).

(ii) False, putting \(\theta=0\) the left hand side is \(1\) and the right hand side is \(\sin(1)\ne 1\).

(iii) False, polynomials are unbounded on the reals while \(|\cos(\theta)| \le 1\).

(iv) True, The function \(f(x)=x^4+3+x^{-4}\) goes to infinity as \(x\) goes to \(0\) from above, and as \(x\) goes to infinity. It also in continuous and differentiable and has one stationary point on \((0,\infty)\). This stationary point therefore must be a global minimum and it occurs at \(x=1\), and \(f(1)=5\).