A fourth degree polynomial function [math]f(x)=a\cdot x^4+b\cdot x^3+c\cdot x^2+d\cdot x+k[/math] has a maximum at [i](1, 10)[/i] and an inflection point at [i](4, -1)[/i]. Furthermore [i]f(x)[/i] has a root at [i]x = -3[/i]. [br]Compute the values for the coefficients [i]a[/i], [i]b[/i], [i]c[/i], [i]d[/i] and [i]k[/i] and plot the graph of [i]f(x)[/i].

[table][tr][td]1.[/td][td]Define the function [i]f(x)[/i] by entering [math]f\left(x\right)=a\cdot x^4+b\cdot x^3+c\cdot x^2+d\cdot x+k[/math] into the [i]Input Bar[/i].[/td][/tr][tr][td]2.[/td][td]To compute the values of the five coefficients, set up five equations using the information given above.[/td][/tr][tr][td][/td][td][i](1, 10)[/i] is a point on the graph of [i]f(x)[/i]: Enter the equation [math]f\left(1\right)=10[/math] into the [i]Input Bar[/i] and press [i]Enter[/i].[/td][/tr][tr][td][/td][td][i](1, 10)[/i] is a maximum of [i]f(x)[/i]: Enter the equation [math]f'\left(1\right)=0[/math] into the [i]Input Bar[/i] and press [i]Enter.[/i][/td][/tr][tr][td][/td][td][i](4, -1)[/i] is a point on the graph of [i]f(x)[/i]: Enter the equation [math]f\left(4\right)=-1[/math] into the [i]Input Bar[/i] and press [i]Enter[/i].[/td][/tr][tr][td][/td][td][i](4, -1)[/i] is an inflection point of [i]f(x)[/i]: Enter the equation [math]f''\left(4\right)=0[/math] into the [i]Input Bar[/i] and press [i]Enter[/i].[/td][/tr][tr][td][/td][td][i](-3, 0)[/i] is a point on the graph of [i]f(x)[/i]: Enter the equation [math]f\left(-3\right)=0[/math] into the [i]Input Bar[/i] and press [i]Enter[/i].[/td][/tr][tr][td]3.[/td][td]Label the five equations by respectively pressing the [i]More[/i] button and selecting [i]Add label[/i].[/td][/tr][tr][td][/td][td][b]Note:[/b] The equations will be labeled [i]eq1[/i] to [i]eq5[/i].[/td][/tr][tr][td]4.[/td][td]Solve the system of equations by entering [math]s=Solve\left(\left\{eq1,eq2,eq3,eq4,eq5\right\}\right)[/math] into the [i]Input Bar[/i].[/td][/tr][/table][table][tr][td]5.[/td][td]Use the command [math]Substitute(f,s)[/math] to apply this solution to the function and to substitute the parameters [i]a[/i] to [i]k[/i] by the computed values. [/td][/tr][tr][td][/td][td][b]Note: [/b]The graph of the function will be displayed in the [i]Graphics View[/i]. [/td][/tr][/table]