Describe the First Derivative Test in your own words.

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Bella · Accepted Answer

The first derivative test is the first step in the process of studying functions to determine their extreme point.
The point is a crucial component of the function. $x = c$ at which either $f (c) = 0$ or $f (c)$ does not exist.
A function is said to have a local (relative) extreme at a crucial point if the derivative of the function changes sign near that location.
At the critical point, the function has a local (relative) maximum if the derivative switches from being positive (growing function) to negative (decreasing function).
The function has a local (relative) minimum at the critical point if the derivative switches from a negative (decreasing function) to a positive (growing function).
The First Derivative Test for Local Extreme Values is the name of the procedure when it is used to find the local maximum or lowest function values.
Step 2
For example,
Let, $f (x) = x^{4} - 8 x^{2}$
Differentiate with respect to x,
$f^{'} (x) = 4 x^{3} - 16 x$
Find critical points by solving $f^{'} (x) = 0$ for x,
$f^{'} (x) = 0$
$4 x^{3} - 16 x - 0$
$4 x (x^{2} - 4) = 0$
$4 x (x - 2) (x + 2) = 0$
$x = 0, x = 2, x = - 2$
Step 3
$f (0) = {(0)}^{4} - 8 {(0)}^{2} = 0$
$f (2) = {(2)}^{4} - 8 {(2)}^{2} = - 16$
$f (- 2) = {(- 2)}^{4} - 8 {(- 2)}^{2} = - 16$
Consequently, the function's key components are (0, 0), (2, -16) and (-2, -16).
The domain of the function $(- \infty, \infty)$ is split into four smaller periods, $(- \infty, - 2), (- 2, 0), (0, 2)$ and $(2, \infty)$
Step 4
$\begin{array}{cc} Interval & Test point & f^{'} (x) & Increasing/Decreasing \\ (- \infty, - 2) & x = - 3 & f^{'} (- 3) = 4 (- 3)^{3} - 16 (- 3) = - 60 < 0 & Decreasing \\ (- 2, 0) & x = - 1 & f^{'} (- 1) = 4 (- 1)^{3} - 16 (- 1) = 12 > 0 & Increasing \\ (0, 2) & x = 1 & f^{'} (1) = 4 (1)^{3} - 16 (1) = - 12 < 0 & Decreasing \\ (2, \infty) & x = 3 & f^{'} (3) = 4 (3)^{3} - 16 (3) = 60 > 0 & Increasing \end{array}$
Because $f^{'} (x)$ changes from negative to positive around −2 and 2, $f (x)$ has a local minimum at (−2, −16) and (2, −16).
Also, $f^{'} (x)$ f has a local maximum at because its value varies from positive to negative at 0. (0, 0).

Describe the First Derivative Test in your own words.

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