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Question:

What is the maximum possible value of (21 Sin X + 72 Cos X)?

Explanation:

Let f (X) = 21 sin X + 72 cos X

∴ f ’(X) = 21 cos X – 72 sin X

At f ’(X) = 0

21 cos X = 72 sin X

∴ tan X =  $\frac{21}{72}$

Drawing the corresponding right triangle we have the following:

​​​​​​​

∴ f ‘’(X) = –21 sin X – 72 cos X

= - 21 × $\frac{7}{25}$ - 72 × $\frac{24}{25}$ < 0

∴ f (X) has a maximum at f ’(X) = 0

∴ Maximum value of f(x) = 21 × $\frac{21}{75}$ + 72 × $\frac{72}{75}$ = 75

Hence, option (d).