Please try again. Recall that a critical number of a function f is a number c in the domain of f such that either f '(c) = 0 or f '(c) does not exist. Use the Quotient Rule to find f '.
Added by Gursimran K.
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To find the critical numbers of a function \( f(x) \) using the Quotient Rule, follow these steps: Show more…
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Critical Numbers Consider the function $$f(x)=\frac{x-4}{x+2}$$ Is $x=-2$ a critical number of $f ?$ Why or why not?
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Extrema on an Interval
Example: f(x) = x^3 – 6x find the relative extrema. Solution: This is the same equation from the last example, so we have the first derivative: f'(x) = 3x^2 – 6 The critical points of f(x) will be where f'(x) = 0, so 0 = 3x^2 – 6 6 = 3x^2 2 = x^2 ± √2 = x From the last example we also already know that f''(–√2) < 0, so we know (–√2, f(–√2)) = (–√2, 4√2) is a relative maximum, and since f''(√2) > 0, we know (√2, f(√2)) = (√2, –4√2) is a relative minimum. Certainly faster than creating another one of those tables, right? Exercise: f(x) = x^3 – 6x^2 – 15x + 45 find the relative extrema. -You are free to use the second derivative test to find relative extrema, as long as you are willing to find the second derivative and the second derivative is not equal to zero at any of the critical points, because again, you can't draw any conclusions in a situation like that. Take for example g(x) = x^4 which has a critical point at x = 0 , but since g''(x) = 12x^2 , g''(0) = 0 , which means we don't know if an extreme exists at x = 0 from the second derivative test. You need to use g'(x) = 4x^3 and the first derivative test to be certain. For the record, a relative minimum exists at (0,0), but I'll let you verify that yourself! -Going back to concavity however, there are several instances of concavity being applied in business and economics that we should go over.
Andrew N.
'Find the critical points of the following function. f(x) = Select the correct choice below and, if necessary; fill in the answer box to complete your choice 0 A The critical point(s) occur(s) at x = (Use a comma to separate answers as needed:) 0 B. There are no critical points_'
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