Answer to Question #297066 in Calculus for Ram

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 $x=0, x=\dfrac{3-\sqrt{3}}{2}, x=\dfrac{3+\sqrt{3}}{2}.$