Optimization Problem

[shouav1]

For the optimization, I have came up with the original function of:

f(x)= 2k √(x²+21²) + k√((45 – x)²+15²)

and f’(x) = (2x) / √(x²+21²) + (2x-90)/(2√((45-x)²+15²)

Now I am suppose to simplify f'(x) to a 4th degree polynomial to before applying Newton's Method.

How do I simplify f’(x) = (2x) / √(x²+21²) + (2x-90)/(2√((45-x)²+15²) to a 4th degree polynomial?

 

[MarkFL]

While it does not affect the outcome, your derivative would actually be:

\(\displaystyle f'(x)=k\left(\dfrac{2x}{\sqrt{x^2+21^2}}+\dfrac{x-45}{\sqrt{(x-45)^2+15^2}} \right)\)

Presumably, we have \(\displaystyle k\ne0\), so to find the critical numbers, we set:

\(\displaystyle \dfrac{2x}{\sqrt{x^2+21^2}}+\dfrac{x-45}{\sqrt{(x-45)^2+15^2}}=0\)

Combine terms on the left:

\(\displaystyle \dfrac{2x\sqrt{(x-45)^2+15^2}-(45-x)\sqrt{x^2+21^2}}{\sqrt{x^2+21^2}\sqrt{(x-45)^2+15^2}}=0\)

Now, since the denominator has no real roots, we are left with:

\(\displaystyle 2x\sqrt{(x-45)^2+15^2}-(45-x)\sqrt{x^2+21^2}=0\)

\(\displaystyle 2x\sqrt{(x-45)^2+15^2}=(45-x)\sqrt{x^2+21^2}\)

Now, squaring both sides will lead to the quartic you desire.