A curve has equation y= e^x -5x, Find the coordinates of the stationary point and show it is a minimum point

differentiating e^x gives e^x (need to know this), differentiating -5x gives -5 (step the power of the x down by 1 and multiply the original power with the coefficient, power of the x was 1 so becomes 0, x⁰ = 1)

so dy/dx = e^x -5, a stationary point is when dy/dx = 0 so to find the x coordinate we say e^x -5 =0, then solve for x

e^x =5

x = ln(5) (we would leave it like this usually)

then put the x value back in to find the y value:  y = e^ln(5) -5 (ln(5))

y = 5 - 5ln(5) (we would leave it in the exact form usually)

so the stationary point is ( ln(5), 5-5ln(5) ) (remember it asked for coordinates)

to prove this is a minimum point we need the second derivative:

differentiating e^x-5 gives e^x, put in x=ln(5) and we get that d²y/dx² = 5 at the stationary point, as this is greater than 0 it is a minimum point.