Determine the critical points of f(x) = 3x^4 - 8x^3 + 6. (2021)

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Question: Determine the critical points of f(x) = 3x^4 - 8x^3 + 6. (2021)

Options:

Correct Answer: (1, 1)

Exam Year: 2021

Solution:

f\'(x) = 12x^3 - 24x^2. Setting f\'(x) = 0 gives x = 0, 2. Check f(1) = 1.

Determine the critical points of f(x) = 3x^4 - 8x^3 + 6. (2021)

Determine the critical points of f(x) = 3x^4 - 8x^3 + 6. (2021)

Step 1: Write down the function f(x) = 3x^4 - 8x^3 + 6.
Step 2: Find the derivative of the function, f'(x). The derivative tells us the slope of the function.
Step 3: Calculate the derivative: f'(x) = 12x^3 - 24x^2.
Step 4: Set the derivative equal to zero to find critical points: 12x^3 - 24x^2 = 0.
Step 5: Factor the equation: 12x^2(x - 2) = 0.
Step 6: Solve for x by setting each factor to zero: 12x^2 = 0 gives x = 0, and x - 2 = 0 gives x = 2.
Step 7: The critical points are x = 0 and x = 2.
Step 8: To check the behavior of the function at these points, you can evaluate f(1) to see the value of the function at x = 1.

Critical Points – Critical points are found by setting the derivative of a function to zero and solving for x.
Derivative Calculation – Finding the first derivative of a polynomial function to analyze its behavior.
Function Evaluation – Evaluating the original function at specific points to determine their nature (maxima, minima, or saddle points).