Question

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.

( f(x)=x^{3}+3 x+2 )

find ( f^{prime}(x) ).

( f(x)=x^{3}+3 x+2 )
( f^{prime}(x)= )

Explanation:

Step1: Apply the power rule

The power rule states that if \(y = x^n\), then \(y^\prime=nx^{n - 1}\). For the function \(f(x)=x^{3}+3x + 2\), the derivative of \(x^{3}\) is \(3x^{2}\) (using \(n = 3\)), the derivative of \(3x\) is \(3\) (using \(n = 1\) for \(x\) and then multiplying by the coefficient \(3\)), and the derivative of a constant \(2\) is \(0\) (since the derivative of a constant \(c\) is \(0\)).

Step2: Combine the derivatives

$$
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Answer:

\(3x^{2}+3\)