Algebra’s strength is in identifying zeros.
This is largely due to the Zero Property Property, which says that
Unless you know of some handy property of a function, our procedure often begin with “get eveything on one side and \(0\) on the other and factor”.
Contrst that to increasing and decreasing, which do not involve equations to solve.
They are comparisons of movement or change between the range and domain.
Our algebra is not that good at such comparisons.
The derivative rephrases this comparison of change back into algebra, where we have
methods.
Let \(f\) be a function.
Then the derivative is denoted as \(f'\).
\(f\) is increasing on intervals where \(f'\) is positive.
\(f\) is decreasing on intervals where \(f'\) is negative.
This allows us to bring our algebraic tools to the question of function behavior.
Any quadratic can be written in the form \(a \, x^2 + b \, x + c\).
The derivative is given by \(2a \, x + b\).
The derivative switches signs at \(\frac {-b}{2a}\), which is the only critical. It corresponds to the first
coordinate of the vertex.
\(\blacktriangleright \) If \(a < 0\), then
- \(f' > 0\) on \(\left ( -\infty , \frac {-b}{2a} \right )\) and \(f\) is increasing on \(\left ( -\infty , \frac {-b}{2a} \right )\).
- \(f' < 0\) on \(\left ( \frac {-b}{2a}, -\infty \right )\) and \(f\) is decreasing on \(\left ( \frac {-b}{2a}, \infty \right )\).
\(\blacktriangleright \) If \(a > 0\), then
- \(f' < 0\) on \(\left ( -\infty , \frac {-b}{2a} \right )\) and \(f\) is decreasing on \(\left ( -\infty , \frac {-b}{2a} \right )\).
- \(f' > 0\) on \(\left ( \frac {-b}{2a}, -\infty \right )\) and \(f\) is increasing on \(\left ( \frac {-b}{2a}, \infty \right )\).
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more examples can be found by following this link
More Examples of the Derivative