Introduction:
To find the maximum value of the given function, we need to find the value of x at which the derivative of the function is equal to zero. This is because at the maximum point, the slope of the function is zero. Let's solve the problem step by step.

Finding the derivative:
To find the derivative of the given function, we need to apply the product rule and the chain rule. The derivative of x^4 is 4x^3, and the derivative of e^(-2x/3) is (-2/3)e^(-2x/3). Applying the product rule, we get:

f'(x) = 4x^3 * e^(-2x/3) + x^4 * (-2/3)e^(-2x/3)

Simplifying the expression, we get:

f'(x) = 4x^3 * e^(-2x/3) - (2/3)x^4 * e^(-2x/3)

Simplifying the derivative:
To find the value of x at which the derivative is equal to zero, we need to solve the equation f'(x) = 0. Let's set the derivative equal to zero and solve for x:

4x^3 * e^(-2x/3) - (2/3)x^4 * e^(-2x/3) = 0

Factoring out common terms, we get:

x^3 * e^(-2x/3) * (4 - (2/3)x) = 0

Since x^3 and e^(-2x/3) are always positive for x > 0, we can ignore them in this equation. So we are left with:

4 - (2/3)x = 0

Solving for x, we get:

(2/3)x = 4

x = 6

Conclusion:
After solving the equation, we find that the value of x at which the derivative is equal to zero is x = 6. Therefore, the function x^4 * e^(-2x/3) has a maximum at x = 6. This means that the function reaches its highest point at x = 6 and starts decreasing thereafter.