Question

The function \( x^4 e^{-2x^2/3} \) (for \( x > 0 \)) has a maximum at a value of \( x \) equal to ...........
(Round off to two decimal places)

Hint:

To find the maximum of a function, differentiate and set the derivative equal to zero. Then solve for \( x \).

Answer 1

Correct Answer: 5.97

Step 1: Differentiating the function.
To find the maximum, we differentiate the function with respect to \( x \) and set the derivative equal to zero. The function is \( f(x) = x^4 e^{-2x^2/3} \). Differentiating: \[ f'(x) = 4x^3 e^{-2x^2/3} - \frac{4}{3} x^5 e^{-2x^2/3} \]

Step 2: Solving for the critical point.
Setting \( f'(x) = 0 \), we get: \[ 4x^3 e^{-2x^2/3} \left( 1 - \frac{x^2}{3} \right) = 0 \] Solving for \( x \), we find that \( x = \sqrt{3} \), which is the location of the maximum.

Step 3: Conclusion.
The value of \( x \) is \( \boxed{1.73} \).