find all critical points of f(x) = x^4 – 4x^3 + 4x^2
A critical point is an interior point of the domain of a function where the derivative is zero or undefined.
Since f(x) is a polynomial, all points of f(x) and f'(x) are defined over ℝ.
Thus it is essential to find all points where f'(x)=0,
4x³-12x²+8x=0
which factors easily to:
4x(x²-3x+2)=0
Can you take it from here?
To find the critical points of a function, we need to find the values of x where the derivative of the function is equal to zero or does not exist.
Step 1: Find the derivative of f(x) with respect to x.
f'(x) = 4x^3 - 12x^2 + 8x
Step 2: Set the derivative equal to zero and solve for x.
4x^3 - 12x^2 + 8x = 0
Step 3: Factor out the common factor, 4x.
4x(x^2 - 3x + 2) = 0
Step 4: Factor the quadratic equation.
4x(x - 2)(x - 1) = 0
Step 5: Set each factor equal to zero and solve for x.
4x = 0 --> x = 0
x - 2 = 0 --> x = 2
x - 1 = 0 --> x = 1
The critical points of f(x) are x = 0, x = 2, and x = 1.