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

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

Correct Answer: x = 0, x = 2

Step 1: Write down the function f(x) = 3x^4 - 8x^3 + 6.
Step 2: Find the derivative of the function, which is 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 equal to zero: 12x^2 = 0 or x - 2 = 0.
Step 7: From 12x^2 = 0, we find x = 0. From x - 2 = 0, we find x = 2.
Step 8: The critical points are x = 0 and x = 2.
Step 9: To find the value of the function at x = 2, calculate f(2) = 3(2^4) - 8(2^3) + 6.
Step 10: Calculate f(2): f(2) = 3(16) - 8(8) + 6 = 48 - 64 + 6 = -10.
Step 11: The critical points are x = 0 and x = 2, with f(2) = -10.

Critical Points – Finding critical points involves calculating the derivative of a function and setting it to zero to find where the function's slope is zero.
Derivative Calculation – Understanding how to differentiate polynomial functions is essential for finding critical points.
Function Evaluation – Evaluating the original function at critical points to determine their nature (e.g., local maxima, minima) is a key step.