Questions: Graph the function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. f(x) = x^(11 / 3) - x^(14 / 3)

Solution Steps

The function \( f(x) = x^{11/3} - x^{14/3} \) is defined for all real numbers \( x \). Therefore, the domain is \( (-\infty, \infty) \).

To find the critical points, we first find the derivative of the function: \[ f'(x) = \frac{11}{3}x^{8/3} - \frac{14}{3}x^{11/3} \] Set \( f'(x) = 0 \) to find the critical points: \[ \frac{11}{3}x^{8/3} - \frac{14}{3}x^{11/3} = 0 \] \[ x^{8/3} \left( \frac{11}{3} - \frac{14}{3}x \right) = 0 \] This gives \( x = 0 \) or \( x = \frac{11}{14} \).

To determine where the function is increasing or decreasing, we analyze the sign of \( f'(x) \):

• For \( x < 0 \), \( f'(x) > 0 \), so the function is increasing.
• For \( 0 < x < \frac{11}{14} \), \( f'(x) > 0 \), so the function is increasing.
• For \( x > \frac{11}{14} \), \( f'(x) < 0 \), so the function is decreasing.

Final Answer