For the curve, y = e^(3x) - 6e^(2x) + 32 ,find the exact x-coordinate of the minimum point and verify that the y-coordinate of the minimum point is 0.

At the minimum point, the gradient will be zero, so we differentiate the equation, giving us dy/dx = 3e^3x - 12e^2x. If the gradient is zero at the minimum point, we can set dy/dx = 0, so 3e^3x - 12e^2x = 0. We rearrange to obtain 3e^3x = 12e^2x , and divide each side by 3 to get e^3x = 4e^2x. We can then divide each side by e^2x to get e^x = 4, and then take the natural log of each side to get x = ln4. To verify that the y co-ordinate is zero, we sub in ln4 for x in the original equation to obtain 64 - 96 +32 =0.