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Use 1st Derivative Test to identify min/max and intervals of increase/decrease of f(x) = x^3 - 6x² + 15.

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Given the following function f(x):


f\mleft(x\mright)=x^3-6x²+15

We will use the 1st Derivative Test to find the min/max and intervals of increase/decrease of f(x).

The 1st derivative will be as follows:


f^(\prime)(x)=3x^2-12x

To find the critical points, we will solve the equation f'(x) = 0


\begin{gathered} 3x^2-12x=0 \\ 3x(x-4)=0 \\ 3x=0\rightarrow x=0 \\ x-4=0\operatorname{\rightarrow}x=4 \end{gathered}

We will identify the points using the 2nd derivative:


\begin{gathered} f^(\prime)^(\prime)(x)=6x-12 \\ x=0\operatorname{\rightarrow}f^(\prime)^(\prime)(0)=-12(-ve)\operatorname{\rightarrow}max. \\ x=4\operatorname{\rightarrow}f^(\prime)^(\prime)(4)=12(+ve)\operatorname{\rightarrow}min. \end{gathered}

So the answer will be:

we have a local maximum at x = 0

And a local minimum at x = 4

The intervals of increase will be: (-∞, 0) ∪ (4, ∞)

The intervals of decrease will be: (1,4)

Use 1st Derivative Test to identify min/max and intervals of increase/decrease of-example-1
User Gaurav Toshniwal
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