Tangents, normals, stationary points, increasing and decreasing functions: แบบฝึกหัด

A curve has the equation \( y = 2x^3 - 9x^2 + 12x - 5 \). Find the set of values of \( x \) for which the function is decreasing.

Find the equation of the normal to the curve \( y = x e^{-x} \) at the point where \( x = 1 \). Give your answer in the form \( y = mx + c \).

The curve \( C \) has equation \( y = x^2 \ln(x) \) for \( x > 0 \). Find the \( x \)-coordinate and the nature of the stationary point of \( C \).

A curve has the equation \( y = 3x^2 - 4x + 1 \).
Find the equation of the normal to the curve at the point where \( x = 1 \). Give your answer in the form \( ax + by + c = 0 \).

Find the equation of the normal to the curve \( y = e^{2x} + 3x \) at the point where \( x = 0 \), giving your answer in the form \( ax + by + c = 0 \).

Consider the function \( f(x) = (x - 2)e^{2x} \).
(a) Using the product rule, find the derivative \( f'(x) \).
(b) Find the exact coordinates of the stationary point of the curve \( y = f(x) \).
(c) By evaluating the sign of \( f'(1) \), determine whether the function is increasing or decreasing at the point where \( x = 1 \).