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Find the critical points of the following function. f(x)=x² √x+12.

User Ahmadux
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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.

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