Describe the First Derivative Test in your own words.

ankarskogC

ankarskogC

Answered question

2021-10-09

Describe the First Derivative Test in your own words.

Answer & Explanation

Bella

Bella

Skilled2021-10-10Added 81 answers

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)=x48x2
Differentiate with respect to x,
f(x)=4x316x
Find critical points by solving f(x)=0 for x,
f(x)=0
4x316x0
4x(x24)=0
4x(x2)(x+2)=0
x=0, x=2, x=2
Step 3
f(0)=(0)48(0)2=0
f(2)=(2)48(2)2=16
f(2)=(2)48(2)2=16
Consequently, the function's key components are (0, 0), (2, -16) and (-2, -16).
The domain of the function (, ) is split into four smaller periods,(, 2), (2, 0), (0, 2) and (2, )
Step 4
IntervalTest pointf(x)Increasing/Decreasing(, 2)x=3f(3)=4(3)316(3)=60<0Decreasing(2, 0)x=1f(1)=4(1)316(1)=12>0Increasing(0, 2)x=1f(1)=4(1)316(1)=12<0Decreasing(2, )x=3f(3)=4(3)316(3)=60>0Increasing
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).

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