# Study the monotony of $(x-1)^2 \times (2-x)^2$

So you have to find the derivative of the function and see where the derivative is negative and where it is positive and you can find in which parts the function is descending and in which parts it is ascending.

I am having problems with this though as I am getting confused during calculations of the derivative (too many "actions"). The result of the derivative I came up with is: $4x^3 - 18x^2 +26x -12$, I don't know how can you find the solutions of this equation though, so I have no idea in which parts the derivative is positive and in which parts is negative. How exactly do you solve this?

• "monotonicity" not "monotony" Oct 30, 2017 at 21:01

You can write the derivative as $2(x-1)(2-x)^2-2(x-1)^2(2-x)=2(x-1)(2-x)((x-1)-(2-x))=2(x-1)(2-x)(2x-3)$
which makes it easy to see where the derivative is zero. As the function is very large when $x$ is large and positive or large and negative, it is decreasing from $-\infty$ to the first point where the derivative is zero, then increasing, and so on.

• Shouldn't it be 2(x-1)(2-x)(-2x+3) ? Anyway thank you for helping. I broke up everything during calculations, apparently i should have been more observant as you could clearly factor the formula. Oct 30, 2017 at 21:01
• No, my calculation and the other one agree on the sign. Also we know the derivative is negative for $x$ large and negative which agrees with my sign rather than yours. Oct 30, 2017 at 21:03

HINT: consider the first derivative in this form $$f'(x)=2 (x-2) (x-1) (2 x-3)$$