Mod, Oh my God!

Algebra Level 4

Let,
$\displaystyle f(x) = x^2-5x+6$
$\displaystyle g(x) = f(|x|)$
$\displaystyle h(x) = |g(x)|$

Find the set of values of $\displaystyle \mu$ , such that the equation $\displaystyle h(x)-\mu=0$ has exactly $\displaystyle 8$ real and distinct roots.

(-\infty,0)

(0,\frac{1}{4})

(-\infty,\frac{1}{4})

(0,\frac{5}{2})

2 solutions

Arghyanil Dey
Apr 26, 2014

$f(x)=x^{2}-5x+6$

$f(x)=(x-2)×(x-3)$

So f(x)=0 is a parabola (upward) which cuts x-axis at point (2,0) & (3,0)

g(x)=f(|x|) , so g(x)=0 is symmetric about y-axis or the the part of f(x)=0 which lies in the right side of y-axis that is reflected on y-axis.

h(x)=|g(x)| , so h(x)=0 is symmetric about x-axis (the whole graph of this function lies above the x-axis.

The function $h(x)-k=0$ has 8 solutions if h(x)=0 and y=k intersect 8 times.

It is possible when the straight line y=k intersects h(x)=0 in the region [x€(2,3) &x€(-2,-3)] [I want to mean that to cut the function h(x)=0 8 times y=k has to cut h(x)=0 in that region]

The maximum value of the function |f(x)|=0 in the region x€(2 3) is 1/4.

So , k€(0, 1/4)

I know that the explanation is incomplete without the graph of the function $h(x)=0$ but I do't know how to attach a graph with this solution.

Can any one help me?

Arghyanil Dey - 7 years, 1 month ago

yeah the best way to solve the problem is to draw the graphs!!!

Sudipan Mallick - 7 years, 1 month ago

Connor Kenway
Apr 27, 2014

just search google for abs(x^2-5abs(x)+6)

Don't worry , it is quite explanatory without the graph. ..

A Brilliant Member - 7 years ago

Mod, Oh my God!

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