[size=85]This applet is an addition to the early [url=https://www.geogebra.org/material/show/id/tfxxakkm]applet[/url]. Here, in addition to calculating the roots of a polynomial equation of degree 5, the points of extrema and inflections are calculated using symbolic formulas.[/size]
[size=85][u][b]Case[/b][/u]: [color=#980000]x0[/color]=1; a₁=1; a₂=1; a₃=-4; a₄=-3; a₅=3; [br][b]Roots[/b] 3: Real- [b]z3[/b], [b]z4[/b], [color=#980000][b]z5[/b][/color] and Complex Conjugate numbers- [b]z1[/b]&[b]z2[/b].[br][b][color=#1e84cc]Stationary points[/color][/b]: 4 ([color=#1e84cc][b]R1,R2,R3,R4[/b][/color]). [color=#1e84cc]r1,[/color] [color=#1e84cc]r2[/color], [color=#1e84cc]r3[/color], [color=#1e84cc]r4[/color] -the roots of the equation [color=#1e84cc]f'(x)=0[/color] are real numbers.[br][b][color=#9900ff]Inflection points[/color][/b]: 3 ( [b][color=#9900ff]I1, I2, I3[/color][/b]). [color=#9900ff]i1[/color], [color=#9900ff]i2[/color],[color=#9900ff] i3[/color]-the roots of the equation f''(x)=0 are real numbers.[/size]
[size=85][b][u]Case[/u][/b]: [color=#980000]x0[/color]=-0.5; a₁=1; a₂=1; a₃=5; a₄=-3; a₅=3; [br][b]Roots[/b] 3: Real- [b]z5[/b] and 2 pairs Complex Conjugate numbers- [b]z1[/b]&[b]z2[/b] and [b]z3[/b]&[b]z4[/b].[br][b][color=#1e84cc]Stationary points [/color][/b]- no. The roots of the equation f'(x): [color=#1e84cc]r1[/color],[color=#1e84cc]r2,r3[/color],[color=#1e84cc]r4[/color] are complex numbers.[br][b][color=#9900ff]Inflection points[/color][/b]: one - [b][color=#9900ff]I1.[/color][/b] The roots of the equation [color=#9900ff]f''(x)[/color]: 1 real ([color=#9900ff]i1[/color]) and two are complex numbers ([color=#9900ff]i2[/color]&[color=#9900ff]i3[/color]).[br][/size][br]
