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The function is $$y=x^2-5x+6$$ I have made $$[f(x_2)-f(x_1)]/(x_2-x_1)$$ It results in $$x_1+x_2-5.$$ What should I do next?

rubik
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prishila
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    Roll it in paper and light it. What is it you want and why? – Zelos Malum Oct 27 '15 at 17:33
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    I don't know what level you are at and what tools are available. Maybe you are supposed to use the fact that $y=(x-2)(x-3)$ is an upward facing parabola, with axis of symmetry the vertical line $x=\frac{2+3}{2}$. Or maybe you are expected to complete the square. You do not mention calculus, but that too works quickly. – André Nicolas Oct 27 '15 at 17:35

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Now suppose that $x_2+h =x_1$ with $ h \to 0$. For a positive monotonicity it has to be $$\lim_{h \to 0 } x_1+x_1+h -5\geq 0$$

Solving for $x_1$

$x_1 \geq \frac{5}{2}$

Therefore for $x \geq \frac{5}{2}$ the function is positive monotonic. And for $x < \frac{5}{2}$ the function is strictly negative monotonic.

callculus42
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