Start with a function for a curved line, such as
f(x) = 1/3x^3 + x^2 + x + 8
Take the first derivative of the continuous function.
f'(x) = x^2 + 2x + 1
Simplify the equation to f'(x) = (x + 1)(x + 1)
Solve for x in the first derivative.
f'(x) = (x + 1)(x + 1)
x = -1
Turn the result into an inequality to determine the sign of the function.
f'(x) > 0 if x < - 1
f'(x) < 0 if x > -1
f'(x) = 0 if x = -1
From negative infinity to -1 the line is increasing, from -1 until positive infinity the line is decreasing and -1 is the inflection point. Therefore, the function is negative, since the inflection point is the highest point.