It is given that f(x) = 2sinhx+3coshx. Show that the curve y = f(x) has a stationary point at x =-½ ln(5) and find the value of y at this point. Solve the equation f(x) = 5, giving your answers exactly

1.Differentiating: f'(x)= 2cosh(x)+3sinh(x) At a stationary point, we know f'(x)=0. Therefore 2cosh(x)+3sinh(x)=0. (easy to forget that unlike nromal trig there is no change in sign) Rearranging gives tanh(x)=-2/3. This can be easily solved using arctanh(x)=1/2ln(1+x/1-x) 2. Writing in terms of exponentials gives 5e^x-e^-x=10 Multiply by e^x. This can then be recognised as a simple quadratic equation in e^x. (sometimes can be awkward to spot)