Derivative math question

It is given that the curve y = (x − 2)e^x has one stationary point.

A. Find the exact coordinates of this point

B. Determine whether this point is a maximum or a minimum point.

y = (x-2)e^x.

Find the first derivative:
y' = (x-2)·(e^x)' + (e^x)·(x-2)'
= (x-2)·e^x + (e^x)·1
= (e^x)(x-2+1)
= (x-1)e^x.

A.
The stationary point is where y' = 0, so 0 = (x-1)e^x, so x = 1 (since e^x ≠ 0).
Substitute this in the original equation: y = (1-2)e^1 = -e. The stationary point is (1,-e).

B.
Find the second derivative:
y'' = (x-1)(e^x)' + (x-1)'e^x
= (x-1)e^x + 1·e^x
= (x-1)e^x + e^x
= e^x(x-1+1)
= xe^x.

If y'' > 0, the point is a minimum. e^x is always > 0, so y'' > 0 if x > 0.
At the stationary point, x = 1, which is > 0, so the point is a minimum.