Question

If M and m denotes maximum and minimum value of $\sqrt{49{\cos }^2\theta +{\sin }^2\theta }+\sqrt{49{\sin }^2\theta +{\cos }^2\theta }$ then find the value of (M+m).

Answer 1

$f\left(x\right)=\sqrt{49co{s}^2\theta +si{n}^2\theta }+\sqrt{49si{n}^2\theta +co{s}^2\theta }\sqrt{1+48co{s}^2\theta }+\sqrt{1+48si{n}^2\theta }\sqrt{1+{48cos}^2\theta +1+{48sin}^2\theta +2\left(\sqrt{1+48si{n}^2\theta +48co{s}^2\theta +48\times 48{\left(sin\theta cos\theta \right)}^2}\right)}\sqrt{50+2\sqrt{1+48+576(sin\theta cos\theta {)}^2}}\sqrt{50+2\sqrt{49+576{sin}^{2}2\theta }}$

At ${sin}^{2}2\theta =1$ ......maximum value

$M=\sqrt{50+2\sqrt{625}}=\sqrt{100}=10$

At $si{n}^{2}2\theta =0$ ......minimum value

$m=\sqrt{50+2\times 7}=\sqrt{64}=8(M+m)=18$