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

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

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

Answer

Explanation:

Step1: Find the derivative

The derivative of $f(x)=x^{4}+3x^{3}-2x^{2}-4x + 5$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=4x^{3}+9x^{2}-4x - 4$.

Step2: Find critical points

Set $f'(x) = 0$, i.e., $4x^{3}+9x^{2}-4x - 4=0$. We can use a graphing calculator (like ALEKS) to find the roots of $f'(x)$. The roots of $f'(x)$ are the critical points.

Step3: Use the second - derivative test

Find the second - derivative $f''(x)=12x^{2}+18x - 4$. Evaluate $f''(x)$ at each critical point. If $f''(c)<0$ where $c$ is a critical point, then $f(x)$ has a local maximum at $x = c$. Using the ALEKS graphing calculator directly to find the local maximum points: The local maximum occurs at approximately $(x,f(x))=(-2.47,16.27)$

Answer:

$(-2.47,16.27)$