Mathematics

This problem is about understanding when a quartic function with no odd-power terms can have a local maximum rather than just a global minimum. The key insight is that $h(x)$ is an even function, so its critical points come in pairs $\pm c$, and you need to analyze the second derivative to determine whether the middle critical point (at $x=0$) is a local max or min. By examining how the coefficient $a$ controls the shape of the parabola formed by $h'(x)=4x(x^2+\frac{a}{2})$, you can find exactly which values of $a$ guarantee a local maximum exists.

This problem is about understanding when a quartic function with no odd-power terms can have a local maximum rather than just a global minimum. The key insight is that $h(x)$ is an even function, so its critical points come in pairs $\pm c$, and you need to analyze the second derivative to determine whether the middle critical point (at $x=0$) is a local max or min. By examining how the coefficient $a$ controls the shape of the parabola formed by $h'(x)=4x(x^2+\frac{a}{2})$, you can find exactly which values of $a$ guarantee a local maximum exists.