Final answer:
To find the critical points of the function f(x) = x²√x+12, we need to find the values of x where the derivative of the function is zero or undefined. The critical points of the function are x = 0, x = -12, and x = -1/4.
Step-by-step explanation:
To find the critical points of the function f(x) = x²√x+12, we need to find the values of x where the derivative of the function is zero or undefined. First, let's find the derivative of f(x).
f'(x) = 2x√x+12 + x²(1/2)(x+12)^(-1/2)
Next, set f'(x) equal to zero and solve for x:
2x√x+12 + x²(1/2)(x+12)^(-1/2) = 0
Since the expression involves a square root, we need to square both sides of the equation to eliminate the square root:
4x²(x+12) + x⁴ = 0
This is a quartic equation, which can be difficult to solve. However, we can simplify it by factoring:
x²(4x+48) + x⁴ = 0
x²(x+12)(4x+1) = 0
Now we have three critical points: x = 0, x = -12, and x = -1/4. These are the values of x where the derivative is zero or undefined.