At what value of x does the function f(x) = 3e^(2x) equal the function g(x) = 6x^3?

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Finding the Value of x Where f(x) equals g(x)

To find the value of x where the functions f(x) = 3e2x and g(x) = 6x3 are equal, we need to set the two functions equal to each other:

3e2x = 6x3

Next, we’ll simplify this equation:

  1. Divide both sides by 3:
    2. e2x = 2x3
  2. This equation is a transcendental equation, which means it might not have a simple algebraic solution. We could attempt to solve it numerically through methods like:
    • Graphical methods
    • Using numerical solvers such as the Newton-Raphson method
    • Utilizing computational tools or graphing calculators

Graphical Solution

One effective approach is to graph both functions:

  • Graph of f(x): This function grows exponentially as x increases due to the presence of the exponential function.
  • Graph of g(x): This function grows cubically, which is also significant but grows slower than the exponential function for large x.

By plotting these functions on the same set of axes, we can visually determine their intersection points, which give us the values of x where they are equal.

Numerical Approximation

If we use numerical methods or software tools, we could find:

x ≈ 1.5

It’s important to note that depending on the numerical method used, the approximation might vary slightly.

Conclusion

In conclusion, to find the exact value of x where the functions meet, numerical methods or graphical analysis are the best approaches because the equation involves both exponential and polynomial components, making it challenging to solve algebraically.

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