Thread: There is a problem from sine & cosine chapter

In ∆ABC, AB = (2 – x) cm, BC = (x + 1) cm and angle ABC = 120˚:

a) Show that AC^2 = x^2 – x +7.
b) Find the value of x for which AC has a minimum value.

I’ve done 1st part. But how can I solve 2nd part. Please help me.

Hello, geton!

$\displaystyle \text{In }\Delta ABC,\;AB \,= \,2 - x,\;BC \,= \,x + 1,\;\angle ABC = 120^o$

a) Show that: .$\displaystyle AC^2 \:= \:x^2 - x +7$

b) Find the value of $\displaystyle x$ for which $\displaystyle AC$ has a minimum value.

You already have: .$\displaystyle AC \:=\:(x^2-x+7)^{\frac{1}{2}}$

Minimize the function: .$\displaystyle AC' \;=\;\frac{1}{2}(x^2-x+7)^{-\frac{1}{2}}(2x-1)$

We have: .$\displaystyle \frac{2x-1}{2\sqrt{x^2-x+7}} \:=\:0\quad\Rightarrow\quad 2x-1 \:=\:0$

Therefore: .$\displaystyle x \:=\:\frac{1}{2}$

Without differentiation, of course

$\displaystyle 2 \cdot \overline {AC} = \sqrt {4x^2 - 4x + 28} = \sqrt {\left( {2x - 1} \right)^2 + 27} .$

The rest follows.

Originally Posted by Krizalid

Without differentiation, of course

$\displaystyle 2 \cdot \overline {AC} = \sqrt {4x^2 - 4x + 28} = \sqrt {\left( {2x - 1} \right)^2 + 27} .$

The rest follows.

You mean $\displaystyle \sqrt {\left( {2x - 1} \right)^2 + 27} = 0$

But in this way how can I get x = 1/2?

Originally Posted by geton

You mean $\displaystyle \sqrt {\left( {2x - 1} \right)^2 + 27} = 0$

But in this way how can I get x = 1/2?

Since$\displaystyle \forall x \in \mathbb{R}, (2x-1)^2 \geq 0$, we have $\displaystyle \sqrt{\left( {2x - 1} \right)^2 + 27} \geq \sqrt{27}$
with equality at 2x-1 = 0.
Thus min is achieved at $\displaystyle x = \frac12$

Thanks.