Questions: Find a) any critical values and b) any relative extrema. g(x)=-x^3+27x-50

Transcript text: Find a) any critical values and b) any relative extrema. \[ g(x)=-x^{3}+27 x-50 \]

Solution

Step 1: Find the derivative of g(x)

The derivative of the function $g(x) = -x^3 + 27x - 50$ is $g'(x) = 27 - 3 x^{2}$.

Step 2: Solve g'(x) = 0 for critical points

Solving $g'(x) = 0$ gives the critical points: -3, 3.

Step 3: Determine the nature of critical points

At $x = -3$, the function has a local minimum.

At $x = 3$, the function has a local maximum.

Step 4: Find relative extrema

At $x = -3$, the function has a value of $g(x) = -1 \cdot 10^{2}$.

At $x = 3$, the function has a value of $g(x) = 4$.

The function has a local minimum at $x = -3$, with $g(x) = -1 \cdot 10^{2}$.

The function has a local maximum at $x = 3$, with $g(x) = 4$.

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