Question

Investigate the existence of a extrema value, what does the markered part mean?

Answer 1

Answer: We have

f'(x) = a x + b,

f'(x) = 0 at x = -b/a

f(x) = a x^2 / 2 + b x + c

Meaning of marked part

❟ ∵ a<0 ❟ f is a quadratic function

∴ f has absolute maximum value at x = -b/a

For all a with a less than zero, f is a quadratic function. Therefore f has a global maximum at x = -b/a

That typesetting seems very sloppy. It probably is supposed to be

∀a < 0, f is a quadratic function.

The second sentence is sloppy in use of "absolute". It can't mean absolute value, so presumably it means "global".

Sometimes a minimum or maximum is only local, but a quadratic function has exactly one extrema, and it is global. And if a < 0, the extrema is a global maximum.

Step-by-step explanation:

An extrema (minimum or maximum) for f(x) occurs only where f'(x) = 0, that is, when the slope of the tangent at x is zero.

But if the function crosses its tangent at that point, the point is an inflection point, not an extrema. A quadratic never crosses it's tangent.