15. Given the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { l l } x ^ { 3 } , & \mathrm { x } \leq \mathrm { a } , \\ \mathrm { x } ^ { 2 } , & \mathrm { x } > \mathrm { a } , \end{array} \right.$ if there exists a real number $b$ such that the function $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) - \mathrm { b }$ has exactly two zeros, then the range of values for $a$ is $\_\_\_\_$. III. Solution Questions: This section has 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps in your answers.
15. Given the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { l l } x ^ { 3 } , & \mathrm { x } \leq \mathrm { a } , \\ \mathrm { x } ^ { 2 } , & \mathrm { x } > \mathrm { a } , \end{array} \right.$ if there exists a real number $b$ such that the function $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) - \mathrm { b }$ has exactly two zeros, then the range of values for $a$ is $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps in your answers.\\