Applications of Derivatives

Algebra Level 5

Find the number of real solutions of the equation $\large {2^{x}|2-|x||=1}.$

6 1 4 3 0 2

1 solution

Kushal Bose
Jul 6, 2016

$|2-x|=2^{-x}$

Case(1) : $x>0$

$\\\\\\\\\\\\$ $|2-x|=2^{-x}$

Subcase (1) $x<2$

$2-x=2^{-x}$

Now ploting the graph og two graphs we can see that one intersection exits.

Subcase(2) $x>2$

Again there exists a one solution

Now Case(2) $x<0$

$\\\\\\\\\\\\$ $|2+x|=2^{-x}$

Subcase(1) $x>-2$ ther exists one solution

Subcase(2) $x<-2$ there is no intersection.

So all total $3$ solutions exist.

Or you may just plot graphs of $2^{-x}$ and $||x|-2|$ and see where they intersect.

Rishabh Jain - 4 years, 11 months ago

Yes sure but you have to consider this cases to draw the graph of ||x|-2|

Kushal Bose - 4 years, 11 months ago

Nope... There are methods of symmetry and transformations... which you can use for quick drawing of graphs of $f(|x|),|f(x)|,f(x+2),$ etc. given graph of $f(x)$ .

Rishabh Jain - 4 years, 11 months ago

@Rishabh Jain – Its right but sometimes you will get more fun by using fundamental rules.because you cannt always remember all formulae.

Kushal Bose - 4 years, 11 months ago

Applications of Derivatives

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