2. Consider a function f : R → R where S(x) = x³ – 3x for all æ e R. Identify all critical points and explain whether each of the critical points is a local maximum, a local minimum, or neither. Are there global maximum or minimum solutions? Explain.
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![**Exercise 2: Analyzing Critical Points of a Function**
Consider a function \( f : \mathbb{R} \to \mathbb{R} \) where
\[ f(x) = x^3 - 3x \quad \text{for all } x \in \mathbb{R}. \]
**Task**
Identify all critical points and explain whether each of the critical points is a local maximum, a local minimum, or neither. Are there global maximum or minimum solutions? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56945fae-aeca-4f60-8c3d-a266c334f238%2Fef7e02b0-fcb7-42dd-a38b-14f848a1d017%2Fbadqsow_processed.png&w=3840&q=75)
Given the function .
To find the critical points and the local maximum and local minimum .
To find this we can use First derivative test.
First derivative test:
Let f be a real valued function continuous at a critical point c () then
1)If changes sign from positive to negative as x increases through c i.e. if at every point sufficiently close to and left c, and at every point sufficiently close to and to the right pf c, then c is a point of local maxima.
2)If changes sign from negative to positive as x increases through c i.e. if at every point sufficiently close to and left c, and at every point sufficiently close to and to the right pf c, then c is a point of local minima.
3)If doesn't exist change sing as x increases through c, then c is neither a point of local maxima nor a local minima. Infact, such a point is called point of inflection.
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