Question

<e> find the turning points and point of inflection on the curve, y=x⁵-5x⁴+5x³-1

Answer 1

$$y'=5x^4-20x^3+15x^2$$

Find the ctitical number(s)

$$y'=0=>5x^4-20x^3+15x^2=0$$

$$5x^2(x^2-4x+3)=0$$

$$5x^2 (x-1)(x-3)=0$$

$$x_1=0, x_2=1, x_3=3$$

Find the second derivative

$$y''=20x^3-60x^2+30x$$

$$y''(0)=0$$

The function $f$ has no turning point at $x=0.$

$$y''(1)=20-60+30=-10<0$$

The function $f$ has turning point at $x=1.$ The function $f$ has a local maximum at $x=1.$

$$y''(3)=540-540+90=90>0$$

The function $f$ has turning point at $x=3.$ The function $f$ has a local minimum at $x=3.$

Find the inflection point(s)

$$y''=0=>20x^3-60x^2+30x=0$$

$$10x(2x^2-6x+3)=0$$

$$5x(2x-3-\sqrt{3})(2x-3+\sqrt{3})$$

Point of inflection at $x=0, x=\dfrac{3-\sqrt{3}}{2}, x=\dfrac{3+\sqrt{3}}{2}.$