Find all critical values of the function \( g(x) = \frac{5}{x^2 - x + 6} \).
Answer (2)
g ( x ) = x 2 − x + 6 5 D : x 2 − x + 6 = 0 Δ = ( − 1 ) 2 − 4 ⋅ 1 ⋅ 6 = 1 − 24 = − 23 < 0 x ∈ R v er t e x o f y = x 2 − x + 6 p = 2 ⋅ 1 − ( − 1 ) = 2 1 q = 4 ⋅ 1 − ( − 23 ) = 4 23 x → 2 1 lim x 2 − x + 6 5 = 4 23 5 = 23 5 ⋅ 4 = 23 20
x → ± ∞ lim x 2 − x + 6 5 = 0
The critical value of the function g ( x ) = x 2 − x + 6 5 is x = 2 1 . This is where the derivative is equal to zero, and the function has no points where it is undefined since the denominator never equals zero. Evaluating the function at this critical point gives g ( 2 1 ) = 23 20 .
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