maximum value of algebraic expression (another)

Question

if $p^2+q^2+r^2=5$ and $p,q,r$ all are real number,

then maximum value of $(4p-3q)^2+(5q-4r)^2+(5p-3r)^2$

what i try . Expanding $(4p-3q)^2+(5q-4r)^2+(5p-3r)^2$

$41p^2+41q^2+25r^2-24pq-40qr-30pr$

$25\times 5+16p^2+16q^2-24pq-40qr-30pr$

How i use inequality to find maximum of given expression

Help me please

Answer 1

We'll prove that $250$ it's a maximal value.

Indeed, $$250\geq(4p-3q)^2+(5q-4r)^2+(5p-3r)^2$$ it's $$50(p^2+q^2+r^2)\geq(4p-3q)^2+(5q-4r)^2+(5p-3r)^2$$ or $$(3p+4q+5r)^2\geq0.$$ The equality occurs for $p^2+q^2+r^2=5$ and $3p+4q+5r=0.$