Find maximum of $|\sqrt{x^4-7x^2-4x+20}-\sqrt{x^4+9x^2+16}|$ without using calculus

The difference between the distances to each of those two points is maximised over the whole plane by any point on the straight line $AB$ that does not lie strictly between $A$ and $B$; and this maximum value is simply $|AB|=2\sqrt{17}$.

And your curve touches this line at $A$ when $x=2$; so the maximum value of the expression is $2\sqrt{17}$, achieved when $x=2$.


By the triangle inequality, for every point $P$, $$||PA|-|PB||\leq |AB|$$ so the maximum is attained when $P=A$.

Tags:

Algebra Precalculus

Maxima Minima

Quadratics

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