Given a fixed point `A(a, b)` and a moving point `P(x, f(x) )` on
the curve `y= f(x)`. Then `AP` will be maximum or minimum if it is
normal to the curve at `P`.


Significance of the Sign of 2nd order Derivative and Points of Inflection :
A point where the graph of function is continuous and has a tangent line and where the concavity changes
is called point of inflection.

# At the point of inflection either `y" = 0` and changes sign or `y"` fails to exist.
# At the point of inflection, the curve crosses its tangent at that point.
# A function can not have point of inflection and extrema at same point.

Note: If `d^2y/dx^2 > 0` then `y` is concave up and if `d^2y/dx^2 < 0` they is concave down.

`y = ax^3 + bx^2 + cx + d`

(1) One real & two imaginary roots. (always monotonic) `AA x in R`

Condition:
`f' (x) ge 0` or `f' (x) le 0` together with either `f' (x) = 0` has no root (i.e. `D < 0`) or `f' (x) = 0` has a root
`x = alpha ` then `f(alpha) = 0`.

(i) either `f'(x) = 0` has no real root
or (ii) if `f'(x) = 0` has a root `x = alpha ` then `f(alpha) = 0`

e.g.

`y = x^3 - 2x^2 + 5x + 4` ` y = (x - 2)^3`
`y ' = 3x^2 - 4x + 5 (D < 0)` ` y' = 3(x - 2)^2 = 0 => x = 2, also f(2) = 0`

Note: In this case if `f' (x) = 0` has a root `x = alpha` and `f(alpha) = 0` this would mean `f (x) = 0` has repeated
roots which is dealt separately.

`f(x _1) *f(x_2) > 0`

where `x_1` & `x_2` are the roots of `f' (x) = 0`

`f(x_1 )* f(x_2) =0`

`f(x_ 1)* f(x_2) < 0`
where `x_1` & `x_2` are the roots of `f ' (x) =0`

`f'(x) ge 0` or `f'(x) le 0` & `f (alpha) = 0`
where `alpha` is a root of `f' (x) = 0`
e.g. `y = (x - 1 )^3`


Note:

(i) Graph of every cubic polynomial must have exactly one point of inflection.

(ii) In case (4) if `f(a)`, `f(b)` , `f(c)` and `f(d)` alternatively change sign.