Question

The least value of ${6tan}^2\varphi +{54cot}^2\varphi +18$ is
I: 54 when $A.M.\ge G.M.$ is applicable for ${6tan}^2\varphi +{54cot}^2\varphi +18$.
II: 54 when $A.M.\ge G.M.$ is applicable for ${6tan}^2\varphi +{54cot}^2\varphi$ and 18 added further.
III: 78 when ${tan}^2\varphi ={cot}^2\varphi$.

• A. I is correct
• B. II are correct
• C. III is correct
• D. none of the above is correct

Answer 1

The correct option is A I is correct

Given equation is

$6ta{n}^2\varphi +54co{t}^2\varphi +18$

As we know $AM\ge GM$

$\frac{6ta{n}^2\varphi +54co{t}^2\varphi +18}{3}\ge (6\times 54\times 18{)}^{\frac{1}{3}}$

$6ta{n}^2\varphi +54co{t}^2\varphi +18\ge 3\times 6\times 3$

$⟹6ta{n}^2\varphi +54co{t}^2\varphi +18\ge 54$

$I$ is correct.

$II:AM\ge GM$

$\frac{6ta{n}^2\varphi +54co{t}^2\varphi }{2}\ge (6\times 54{)}^{\frac{1}{2}}$

$6ta{n}^2\varphi +54co{t}^2\varphi \ge 2\times 6\times 3$

Adding 18 On both sides

$6ta{n}^2\varphi +54co{t}^2\varphi +18\ge 36+18\ge 54$

$II$ is correct.