Given is the function [math]f(x)=\sqrt{x}\left(x^2-10x+25\right)[/math].[br]a) Plot the graph of [i]f(x)[/i].[br]b) Calculate the roots, local extrema and inflection point of [i]f(x)[/i].[br]c) Estimate the arguments [i]x[/i], where the tangent on [i]f(x)[/i] has a slope of 30°.[br]d) Estimate the equation of the tangent on the graph of [i]f(x)[/i] in [i]x = 2[/i].

[table][tr][td]1.[/td][td]Enter the equation of the function [math]f\left(x\right)=\sqrt{x}\left(x^2-10x+25\right)[/math] into the [i]Input Bar[/i] and press [i]Enter[/i].[/td][/tr][tr][td][br][/td][td][b]Note:[/b] The graph of [i]f(x)[/i] will be displayed in the [i]Graphics View.[/i][/td][/tr][tr][td]2.[/td][td]Calculate the roots of [i]f(x) [/i]by entering the command [math]Solve(f=0)[/math] or [math]Root(f)[/math] into the [i]Input Bar[/i].[/td][/tr][tr][td]3.[/td][td]To calculate the local extrema of [i]f(x)[/i] use the command [math]Solve\left(f'\left(x\right)=0\right)[/math].[/td][/tr][tr][td]4.[/td][td]Check if [i]x = 1[/i] and [i]x = 5[/i] are arguments of minimal or maximal turning points by calculating the second derivative [math]f''(1)[/math] and [math]f''(5)[/math].[/td][/tr][/table][table][tr][td]5.[/td][td]Calculate the y-coordinates of the turning points by entering [math]f\left(\left\{1,5\right\}\right)[/math] into the [i]Input Bar[/i].[/td][/tr][/table]

[table][tr][td]6.[/td][td]To calculate the inflection points, use the command [math]Solutions\left(f''\left(x\right)=0\right)[/math] and choose [i]Add label [/i]from the context menu to name the list of solutions [i]l1[/i].[br]As only one of the solutions is in the domain of [i]f(x)[/i], enter [math]a=Element\left(l1,1\right)[/math] to label this solution and to be able to reuse it in further calculations.[br][/td][/tr][tr][td]7.[/td][td]Calculate the y-coordinate of the inflection point by entering [math]b=f\left(a\right)[/math] into the [i]Input Bar. [br][/i]You can now display the inflection point by entering its coordinates[i] A=(a, b).[/i][/td][/tr][tr][td]8.[/td][td]To find the arguments [i]x[/i], where f(x) has a slope of 30°, enter the command [math]Solve\left(f'\left(x\right)=tan\left(30°\right)\right)[/math].[/td][/tr][tr][td]9.[/td][td]Estimate the equation of the tangent on [i]f(x)[/i] at [i]x = 2 [/i]by entering [math]Tangent(2,f)[/math] into the [i]Input Bar. [/i]The tangent will be displayed in the [i]Graphics View[/i].[br][/td][/tr][/table]