Question

Find the range of the function f(x) =

${27}^{sin5x} \times {81}^{cos5x}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: f(x) = {27}^{sin5x} \times {81}^{cos5x}$

can be rewritten as

$\rm \: f(x) = {( {3}^{3})}^{sin5x} \times {( {3}^{4})}^{cos5x}$

$\rm \: f(x) = {3}^{3sin5x} \times {3}^{4cos5x}$

$\rm \: f(x) = {3}^{3sin5x + 4cos5x}$

Now, Consider

$\rm \: 3sin5x + 4cos5x$

Now, multiply and divide by square root of the [ square of coefficient of sin5x plus square of coefficient of cos5x ].

So,

$\rm \: Multiply \: and \: divide \: by \: \sqrt{ {3}^{2} + {4}^{2} } = \sqrt{9 + 16} = 5$

So, above expression can be rewritten as

$\rm \: = \: 5\bigg( \dfrac{3}{5} sin5x + \dfrac{4}{5} cos5x\bigg)$

Let assume that

$\rm \: cosy = \dfrac{3}{5}$

$\rm \: cos^{2} y = \dfrac{9}{25}$

$\rm \: 1 - sin^{2} y = \dfrac{9}{25}$

$\rm \: sin^{2} y =1 - \dfrac{9}{25}$

$\rm \: sin^{2} y = \dfrac{25 - 9}{25}$

$\rm \: sin^{2} y = \dfrac{16}{25}$

$\rm\implies \: \: siny = \dfrac{4}{5}$

So, above expression can be rewritten as

$\rm \: = \: 5(sin5x \: cosy \: + \: siny \: cos5x)$

$\rm \: = \: 5sin(5x + y)$

We know,

$\rm \: - 1 \: \leqslant \: sin(5x + y) \: \leqslant \: 1$

$\rm \: - 5 \: \leqslant \: 5sin(5x + y) \: \leqslant \: 5$

$\rm\implies \:\rm \: - 5 \: \leqslant \: 3sin5x + 4cos5x \: \leqslant \: 5$

So,

$\rm\implies \: {3}^{ - 5} \leqslant {3}^{3sin5x + 4cos5x} \leqslant {3}^{5}$

$\rm\implies \: {3}^{ - 5} \leqslant f(x) \leqslant {3}^{5}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

ADDITIONAL INFORMATION

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Range \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf asinx + bcosx & \sf [ - \sqrt{ {a}^{2} + {b}^{2} }, \: \: \sqrt{ {a}^{2} + {b}^{2} }] \\ \\ \sf asinx + bcosx + c & \sf [c - \sqrt{ {a}^{2} + {b}^{2} }, \: c + \sqrt{ {a}^{2} + {b}^{2} }] \end{array}} \\ \end{gathered} \\ \end{gathered}$

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Range \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf y = sinx & \sf   - 1 \leqslant y \leqslant 1\\ \\ \sf y = cosx & \sf  - 1 \leqslant y \leqslant 1 \\ \\ \sf y = tanx & \sf y \:  \in \: ( -  \infty , \infty )\\ \\ \sf y = cosec & \sf y \leqslant  - 1 \:  \: or \:  \: y \geqslant 1\\ \\ \sf y = secx & \sf y \leqslant  - 1 \:  \: or \:  \: y \geqslant 1\\ \\ \sf y = cotx & \sf y \:  \in \: ( -  \infty , \infty ) \end{array}} \\ \end{gathered}$