\( g(x)=-x^{3}+3 x \) (a) Determine whether \( g \) is even, oddd, or neither. (b) There is a local mininum of -2 at -1 . Determine the local maximum. (a) Determine whether \( g \) is even, odd, or neither. Odd Even Neither

To determine the nature of the function \( g(x) = -x^3 + 3x \), we can test if it’s even, odd, or neither by checking the property of \( g(-x) \). When we substitute \(-x\) into the function, we get: \[ g(-x) = -(-x)^3 + 3(-x) = x^3 - 3x. \] Since \( g(-x) = -g(x) \), the function \( g(x) \) is classified as odd. Now for the local maximum! Since \( g(x) \) is a cubic polynomial, it will have a single local maximum and a single local minimum. Given that there’s a local minimum at \( x = -1 \) with value -2, you can find the local maximum by looking at the critical points. By finding the derivative and setting \( g'(x) = 0 \), we can solve for the x-values of critical points. A quick analysis will give you the local maximum between the two critical points, leading you to its specific value!