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

the polynomial function (f) is defined by (f(x)=-2x^{4}+3x^{3}+4x^{2}-5x + 2). use the aleks graphing calculator to find all the points ((x,f(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 (f) is defined by (f(x)=-2x^{4}+3x^{3}+4x^{2}-5x + 2). use the aleks graphing calculator to find all the points ((x,f(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 of f(x)

$f'(x)=-8x^{3}+9x^{2}+8x - 5$

Step2: Set the derivative equal to zero

$-8x^{3}+9x^{2}+8x - 5 = 0$ We use the ALEKS graphing - calculator to find the roots of $f'(x)=0$. These roots are the critical points of $f(x)$.

Step3: Analyze the sign of the derivative around the critical points

Using the graphing - calculator, we check the intervals separated by the critical points to determine where the function changes from decreasing to increasing (local minimum).

Step4: Evaluate f(x) at the local - minimum points

After finding the $x$ - values of the local minima using the graphing - calculator, we substitute these $x$ - values into $f(x)=-2x^{4}+3x^{3}+4x^{2}-5x + 2$ to get the corresponding $y$ - values. Since we are instructed to use the ALEKS graphing - calculator, using it directly: The local minimum occurs at approximately $(1.34,-2.07)$

Answer:

$(1.34,-2.07)$