Questions: The function f(x)=2x^3-36x^2+210x-7 has two critical numbers. The smaller one is x= and the larger one is x=

Transcript text: The function $f(x)=2 x^{3}-36 x^{2}+210 x-7$ has two critical numbers. The smaller one is $x=$ $\square$ and the larger one is $x=$ $\square$

Solution

Solution Steps

Step 1: Find the Derivative

To find the critical numbers of the function $ f(x) = 2x^3 - 36x^2 + 210x - 7 $, we first calculate its derivative:

\[ f'(x) = 6x^2 - 72x + 210 \]

Step 2: Set the Derivative to Zero

Next, we set the derivative equal to zero to find the critical points:

\[ 6x^2 - 72x + 210 = 0 \]

Step 3: Solve for Critical Points

By solving the quadratic equation, we find the critical points:

\[ x = 5 \quad \text{and} \quad x = 7 \]

Final Answer

The smaller critical number is $ \boxed{x = 5} $ and the larger critical number is $ \boxed{x = 7} $.

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Questions: The function f(x)=2x^3-36x^2+210x-7 has two critical numbers. The smaller one is x= and the larger one is x=

Solution

Solution Steps

Final Answer

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