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For what real value of $$N$$ does the range of the function $$y = f(x) = \frac{4x^2 + Nx + N}{x + 1},$$, where $$x$$ is real (and $$x \neq -1$$) consist of all real numbers except for a single interval of the form $$-L < y < L$$?

Apr 6, 2019

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$$\dfrac{df}{dx} = 0 \Rightarrow x = -2,~0\\ \text{we want } f(0) = -f(-2) = L\\ f(0) = N\\ f(-2) = N-16\\ N = 16-N\\ N=8$$

$$f(x) \in \mathbb{R} - (-8,8)$$

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Apr 6, 2019