2x*sqrt(3x^2+4)

Find all critical numbers:

To find the critical numbers of the function f(x) = 2x*sqrt(3x^2+4), we need to first determine where the derivative of the function is equal to zero or undefined. Here's how to do it:

Step 1: Find the derivative of the function.
f(x) = 2x * sqrt(3x^2 + 4)
To find the derivative, we can use the product rule and the chain rule.

First, differentiate the term 2x:
f'(x) = 2 * sqrt(3x^2 + 4) + 2x * (1 / (2 * sqrt(3x^2 + 4))) * (6x)

Simplifying this expression, we get:
f'(x) = 2 * sqrt(3x^2 + 4) + 6x^2 / sqrt(3x^2 + 4)

Step 2: Set the derivative equal to zero and solve for x.
2 * sqrt(3x^2 + 4) + 6x^2 / sqrt(3x^2 + 4) = 0

To solve this equation, we need to isolate the radical term and then square both sides of the equation:
2 * sqrt(3x^2 + 4) = -6x^2 / sqrt(3x^2 + 4)

Now, square both sides:
(2 * sqrt(3x^2 + 4))^2 = (-6x^2 / sqrt(3x^2 + 4))^2

4 * (3x^2 + 4) = 36x^4 / (3x^2 + 4)

Simplifying this expression, we get:
12x^2 + 16 = 36x^4 / (3x^2 + 4)

Multiply both sides by (3x^2 + 4):
12x^2 * (3x^2 + 4) + 16 * (3x^2 + 4) = 36x^4

Simplifying this expression, we get a quartic equation:
36x^4 + 48x^2 + 48x^2 + 64 = 36x^4

Combining like terms and simplifying:
0 = 72x^2 + 64

Step 3: Solve the quartic equation for x.
To solve the quadratic equation, let's set it equal to zero:
72x^2 + 64 = 0

Now, solve for x by factoring or using the quadratic formula.

Factoring:
8(9x^2 + 8) = 0

Setting each factor equal to zero:
9x^2 + 8 = 0 and 8 = 0 (no solutions)

9x^2 = -8

Taking the square root of both sides:
3x = ±√(-8)

Since we cannot take the square root of a negative number to solve for x, there are no critical numbers in this case.

Therefore, the function f(x) = 2x*sqrt(3x^2+4) has no critical numbers.