Critical points
Refer to PennCalc Main/Optimization
For analyzing a function, it's very efficient to have a look at its Critical points
, which could be classified as Extrema
, Inflection
, Corner
, and Discontinuity
.
How to find critical points
How to find critical points
Strategy:
Knowing that
f(x)
has critical pointc
whenf'(c) = 0
orf'(c) is undefined
Differentiate
f(x)
to getf'(x)
Solve
c
forf'(c)=0 & f'(c) undefined
[Refer to Symbolab's step-by-step solution.](https://www.symbolab.com/solver/step-by-step/critical points%2C f\left(x\right)%3Dx\cdot sqrt\left(4-x\right))
Example
See that original function
f(x)
is undefined atx = 2 or -2
Differentiate
f(x)
to getf'(x)
:Solve
f'(x)=0
only whenx=0
.f'(x)
is undefined whenx=2 or -2
, as the same withf(x)
so it's not a solution.
Example
Differentiate
f(x)
to getf'(x)
:f'(x)
is undefined whenx > 4
Solve
f'(x)=0
getx = 8/3
So under the given condition, only
x=8/3
is the answer.
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