Calculus Suppose y=f(x) is a curve that always lies above the x-axis and never has a horizontal tangent, where

Question

Suppose y=f(x) is a curve that always lies above the x-axis and never has a horizontal tangent, where f is differentiable everywhere. For what value of y is the rate of change of y^3 with respect to x seventy-five times the rate of change of y with respect to x?

Answer 1

Sounds like y = e^x, or e^x + 1, or some such. Let's go with e^x. Then y^3 = e^(3x), and rate of change is dy^3/dx = 3e^(3x), so

3e^3x = 75e^x
e^2x = 25
2x = ln 25
x = (1/2) ln 25 = ln 5, and
y = e^(ln 5) = 5

Answer 2

If it never has a horizontal tangent that means you never have a zero derivative. Ie. dy/dx ≠ 0, so that you may safely divide by dy/dx.

3y² * (dy/dx) = dy³/dx = 75 * dy/dx
Since dy/dx ≠ 0, we divide through by it:
3y² = 75
Thus
y = 5, since it lies above the x axis.