Challenging Questions Mathematics – Compulsory Part Learning Unit: More about trigonometry Question M0-14-01 Given that \(\cos{x} = \frac{\sqrt{1+\sin{x}}-\sqrt{1-\sin{x}}}{2}\), find \(\tan{x}\) # More challenging questions will be updated # Answer
Problem Statement: We are given the functions \[f(x)=e^x−ax\] and \[g(x)=ax−\ln x\] with \(x>0\) for \(g(x)\) It is given that \(f(x)\) and \(g(x)\) have the same minimum value. (1) Find \(a\) (2) Prove that there exists a line \(y=b\) intersecting both curves \(y=f(x)\) and \(y=g(x)\) at three distinct points whose \(x\)-coordinates form an arithmetic progression Answer: (1) \(f'(x)=e^x-a\) \(f′(x)=0\) ⟹ \(0=e^x-a\) ∴ \(x= \ln a\) The minimum value... » read more