Math questions truning point and point of inflexion:?

Question

Draw neat sketches of the following functions showing truning points and points of inflexion: please draw it too.

1) y=x^3 - 6x^2 + 9x - 5

2) y= x^4 - 8x^2 + 16

3) y= 1 / ( (x + 1)(x-2)

Answer 1

in order to find inflection point(s) , find the value(s) of x for which d²y/dx² = 0

suppose for  x = a  , d²y/dx²  = 0

In order for x = a to be an inflection point, d²y/dx² must change signs.

When x < a, (i) f''(x) > 0 or (ii)f''(x) < 0 (You can find this out by testing a number less than a in the second derivative).

When x > a, (i) f''(x) < 0 or (ii) f''(x) > 0(again you find this out by testing a number greater than a).

Since f''(x) changes signs at x = a, then (a, y) is an inflection point.