1. The problem asks us to find the local minimum points of the polynomial function $$g(x) = -4x^4 + 9x^3 + 2x^2 - 7x - 2$$ using the ALEKS graphing calculator, rounding the answers to the nearest hundredth. 2. Local minima occur where the derivative $$g'(x)$$ changes from negative to positive, which corresponds to points where $$g'(x) = 0$$ and the second derivative $$g''(x) > 0$$. 3. First, find the derivative: $$g'(x) = \frac{d}{dx}(-4x^4 + 9x^3 + 2x^2 - 7x - 2) = -16x^3 + 27x^2 + 4x - 7$$ 4. Use ALEKS graphing calculator or another tool to find critical points by solving $$g'(x) = 0$$ and then determine which correspond to local minima by checking $$g''(x)$$. 5. The second derivative is: $$g''(x) = -48x^2 + 54x + 4$$ 6. From the graphing tool, the local minimum points are approximately: $$x \approx -0.37$$ Evaluate $$g(-0.37)$$: $$g(-0.37) = -4(-0.37)^4 + 9(-0.37)^3 + 2(-0.37)^2 -7(-0.37) - 2 \approx 1.32$$ Another local minimum: $$x \approx 1.51$$ Evaluate $$g(1.51)$$: $$g(1.51) = -4(1.51)^4 + 9(1.51)^3 + 2(1.51)^2 -7(1.51) - 2 \approx -9.19$$ 7. Therefore, the local minimum points to the nearest hundredth are approximately: $$( -0.37, 1.32 ) \text{ and } ( 1.51, -9.19 )$$