The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Practice Questions
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Q1
The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
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Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Questions & Step-by-step Solutions
1 item
Q
Q: The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Solution: Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Steps: 9
Step 1: Write down the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the first derivative f'(x) by differentiating f(x). This gives f'(x) = 3x^2 - 12x + 9.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 12x + 9 = 0.
Step 4: Simplify the equation by dividing everything by 3: x^2 - 4x + 3 = 0.
