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

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

e=5

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

then put the x value back in to find the y value:  y = eln(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 ex-5 gives ex, 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.

JC
Answered by James C. Maths tutor

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