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For the function $y=(x-1)(x-2)(x-3)(x-4)$, I see graphically that the range is $\ge-1$. But I cannot find a way to determine the range algebraically?

Bob Simons
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    It's a positive quartic with 4 roots, so you just need to find the 2 minima. However you have marked your question as "precalculus" so perhaps doing this graphically is the best option. Otherwise I would suggest differentiating. – Joffan May 23 '15 at 16:32
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    The word "analytically" usually means by calculus. Are calculus means acceptable to you? – Rory Daulton May 23 '15 at 16:33

1 Answers1

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Non-calculus Approach/ Completing Square Approach

Let $u=(x-2)(x-3)=x^2-5x+6$

$(x-1)(x-4)=x^2-5x+4=u-2$

So \begin{align} y&=(x-1)(x-2)(x-3)(x-4)\\&=u(u-2)\\&=u^2-2u\\&=(u-1)^2-1 \end{align}

As $(u-1)^2\ge0$ for all $u\in{\Bbb{R}}$

$y\ge0-1=-1$

Mythomorphic
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