A curve C with an equation y = sin(x)/e^(2x) , 0<x<pi has a stationary point at P. Find the coordinates ofP?

Firstly notice that to find a stationary point of a curve with an equation, the best first step is often to differentiate the expression. Then solve the equation for x when dy/dx = 0. In this case the y equation is a fraction therefore the best approach is to use the quotient rule; if y = u/v then dy/dx = (vdu/dx - udv/dx)/v² given in the formula booklet (if not learn). Therefore in this instanceu = sin x and v = e^2x, from known differentiable functions du/dx = cos x and dv/dx = 2e^2x . Then we substitute these values into are formulas to get; dy/dx = (e^2xcos(x) - 2e^2xsin(x))/(e^2x)² The next step is to do some algebra to solve 0 = dy/dx = (e^2xcos(x) - 2e^2xsin(x))/(e^2x)². 0 = (e^2xcos(x) - 2e^2xsin(x))/(e^2x)²= (e^2xcos(x) - 2e^2xsin(x))/(e^4x) using basic law of indices. = (e^2x)(cos(x) - 2sin(x))/(e^4x) = (e^-2x)(cos(x) - 2sin(x)) = 0 factoring out e^2x and then diving by the e^4xTherefore the solutions to this equation is e^-2x = 0 = cos(x) - 2sin(x), as e^-2x has no solutions therefore the only possible solution is 0 = cos(x) - 2sin(x), the is solve by taking 2sin(x) to the other side and then dividing by sin(x), 2sin(x) = cos(x) then 1/2 = tan(x)therefore the x value at P is x = arctan(1/2) = 0.464 (3.s.f) then to find the y coordinate simply put this value of x into the original equation using the answer function on the calculator to keep the answer accurate to find y = 0.177 (3.s.f). Hence the coordinates at P are (0.464,0.177) to 3.s.f