$f(x)=\sqrt{(x-1)^2+(x^2-5)^2}\;\;,\;\; g(x)=\sqrt{(x+2)^2+(x^2+1)^2},\forall x\in \mathbb{R}$, Then Max $\left\{f(x)-g(x)\right\}$

Question

Let $f(x)=\sqrt{(x-1)^2+(x^2-5)^2}\;\;,\;\; g(x)=\sqrt{(x+2)^2+(x^2+1)^2},\forall x\in \mathbb{R}$.

Find the Minimum of function $\left\{f(x)+g(x)\right\}$ and the maximum of function $\left\{f(x)-g(x)\right\}$.

$\bf{My\; Try}$:: For Minimum of $\left\{f(x)+g(x)\right\}$

Using Minkowski inequality or $\triangle$ Inequality $\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\geq \sqrt{(a+b)^2+(c+d)^2}$

and equality holds when $\displaystyle \frac{a}{b} = \frac{c}{d}$

$\sqrt{(1-x)^2+(5-x^2)^2} + \sqrt{(x+2)^2+(x^2+1)^2}$

$\geq \sqrt{\left(1-x+x+2\right)^2+\left(5-x^2+x^2+1\right)^2} = \sqrt{3^2+6^2} = 3\sqrt{5}$

and equality hold, when $\displaystyle \frac{1-x}{5-x^2} = \frac{x+2}{x^2+1}$

But I did not understand How can I calculate Maximum of $\left\{f(x)-g(x)\right\}$

Help Required

Thanks

Answer 1

Use the same inequality again, but backwards:

$$\sqrt{(x+2)^2 + (x^2+1)^2} + \sqrt{((x-1)-(x+2))^2 + ((x^2-5)-(x^2+1))^2} \geq \sqrt{(x-1)^2 + (x^2-5)^2}$$

This gives us $f(x)-g(x) \leq 3\sqrt{5}$. It also allows for us to solve for $x$.

One warning: this method is really finding the maximum value of $|f(x)-g(x)|$. So when we solve for $x$, we will get two solutions, one maximizing $f(x)-g(x)$, the other maximizing $g(x)-f(x)$.

Answer 2

If one observes carefully $f(x)$ is nothing but distance of the point $A(1,5)$ from an arbitrary point $P(x,x^2)$ lying on the curve $y=x^2$.

Similarly $g(x)$ is nothing but distance of the point $B(-2,-1)$ from any arbitrary point $P(x,x^2)$ on the curve $y=x^2$.

Now,$AB=3\sqrt{5}$

In other words we have been asked to find maximum value of $\mid PA-PB\mid$ and the minimum value of $ PA+PB$. By triangle inequality $\mid PA-PB\mid\leq AB$ and $PA+PB\geq AB$.

Equality occurs when the points $A,P,B$ are collinear which we see happening when $x=-1,3$.

When $x=-1$, then $P(-1,1)$ lies between $A$ and $B$. Here $PA=2\sqrt{5},PB=\sqrt{5}$

Thus $\left( PA+PB\right)_{min}=AB=3\sqrt{5}$

When $x=3$, then $P(3,9)$ does not lie between $A$ and $B$.Here $PA=2\sqrt{5},PB=5\sqrt{5}$

Thus $\mid PA-PB\mid_{max}= AB=3\sqrt{5}$