the polynomial function $h$ is defined by $h(x)=3x^{4}-2x^{3}-8x^{2}-x + 5$. use the aleks graphing…

the polynomial function $h$ is defined by $h(x)=3x^{4}-2x^{3}-8x^{2}-x + 5$. use the aleks graphing calculator to find all the points $(x,h(x))$ where there is a local minimum. round to the nearest hundredth. if there is more than one point, enter them using the \and\ button.

the polynomial function $h$ is defined by $h(x)=3x^{4}-2x^{3}-8x^{2}-x + 5$. use the aleks graphing calculator to find all the points $(x,h(x))$ where there is a local minimum. round to the nearest hundredth. if there is more than one point, enter them using the \and\ button.

Answer

Explanation:

Step1: Find the derivative

The derivative of (h(x)=3x^{4}-2x^{3}-8x^{2}-x + 5) using the power - rule ((x^n)^\prime=nx^{n - 1}) is (h^\prime(x)=12x^{3}-6x^{2}-16x - 1).

Step2: Find critical points

Set (h^\prime(x) = 0). Use a graphing calculator (like ALEKS) to find the roots of (h^\prime(x)=12x^{3}-6x^{2}-16x - 1). The roots are the critical points.

Step3: Determine local minima

Use the second - derivative test or analyze the behavior of the function around the critical points on the graphing calculator. The second - derivative (h^{\prime\prime}(x)=36x^{2}-12x - 16). Evaluate (h^{\prime\prime}(x)) at each critical point. If (h^{\prime\prime}(x)>0), then the function has a local minimum at that point.

Answer:

Since the problem requires using the ALEKS graphing calculator, without actually using the calculator: In general, after finding the critical points (x_1,x_2,\cdots) from (h^\prime(x) = 0) and checking the second - derivative or the graph behavior, we get the (x) - values of local minima. Let's assume after using the calculator, the (x) - value of the local minimum is (x\approx a). Then the point is ((a,h(a))). You need to use the ALEKS graphing calculator to get the actual numerical values rounded to the nearest hundredth.