Math Problem Statement
Find the critical numbers for the given functions: a) f(x) = x^3 - 6x^2 + 9x + 10, b) f(x) = e^x / (x - 2), c) f(x) = 2√(x(6 - x))
Solution
The problem asks to find the critical numbers for the given functions:
Part a:
- Find the derivative of :
- Set the derivative equal to zero to find the critical points: Divide the equation by 3: Factor the quadratic: So, the critical points are and .
Part b:
- Find the derivative using the quotient rule:
- Set the derivative equal to zero to find the critical points: Since , the numerator , so: However, we must also check the denominator for restrictions. makes the denominator zero, so it's a vertical asymptote and not a critical point. Therefore, the only critical number is .
Part c:
- Rewrite the function in a more manageable form:
- Find the derivative using the chain rule: Simplifying the derivative:
- Set the derivative equal to zero: So, , which gives:
- Check the domain: The function is defined where , or , which implies .
Thus, the critical point is .
Would you like more details or have any questions?
Here are 5 related questions for practice:
- What are the critical points for ?
- How do you apply the quotient rule to a more complex function like ?
- Can a critical point also be an inflection point?
- How do you determine if a critical point is a minimum, maximum, or neither?
- How can you apply implicit differentiation to find critical points of implicit functions?
Tip: Critical numbers occur where the derivative is zero or undefined, but make sure to check the domain of the function!
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Critical Points
Quotient Rule
Chain Rule
Formulas
f'(x) = 3x^2 - 12x + 9 for function a
Quotient rule: f'(x) = [(x - 2)(e^x) - (e^x)(1)] / (x - 2)^2 for function b
f'(x) = (6 - 2x) / √(6x - x^2) for function c
Theorems
Critical points occur where f'(x) = 0 or f'(x) is undefined
Quotient Rule
Chain Rule
Suitable Grade Level
Grades 10-12