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

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

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

Answer

Explanation:

Step1: Graph the function

Use the ALEKS graphing calculator to plot ( y = 3x^{4}+5x^{3}-4x^{2}-5x - 2).

Step2: Identify local maxima

A local maximum is a point where the function changes from increasing to decreasing. By observing the graph of the polynomial function (y = 3x^{4}+5x^{3}-4x^{2}-5x - 2) on the graphing calculator: The (x) - coordinate of the local maximum is found by looking at the peak of the relevant part of the graph. After using the graphing - calculator's features (such as the "maximum" function in the graphing utility), we find that (x\approx - 1.17). Substitute (x=-1.17) into the function (f(x)=3x^{4}+5x^{3}-4x^{2}-5x - 2): [ \begin{align*} f(-1.17)&=3(-1.17)^{4}+5(-1.17)^{3}-4(-1.17)^{2}-5(-1.17)-2\ &=3\times(1.17)^{4}-5\times(1.17)^{3}-4\times(1.17)^{2}+5\times1.17 - 2\ \end{align*} ] ((1.17)^{4}=1.17\times1.17\times1.17\times1.17\approx1.87), ((1.17)^{3}=1.17\times1.17\times1.17\approx1.60), ((1.17)^{2}=1.37) [ \begin{align*} f(-1.17)&=3\times1.87-5\times1.60 - 4\times1.37+5\times1.17-2\ &=5.61-8.0 - 5.48 + 5.85-2\ &=(5.61 + 5.85)-(8.0 + 5.48+2)\ &=11.46 - 15.48\ &=- 4.02 \end{align*} ]

Answer:

((x,f(x))=(-1.17,-4.02))