Question

Find the maximum and minimum value of

$f(x) = 11 - 3 {sin}^{2}x + 5 {cos}^{2}x$
​

Answer 1

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

Given function is

$\rm \: f(x) = 11 - {3sin}^{2}x + {5cos}^{2}x$

In order to find the maximum and minimum value, let first reduce the given expression in terms of one Trigonometric function.

We know,

$\boxed{\tt{ \: 1 + cos2x = {2cos}^{2}x \: }}$

and

$\boxed{\tt{ \: 1 - cos2x = {2sin}^{2}x \: }}$

So, using these results, we get

$\rm \: f(x) = 11 - \dfrac{3}{2}(1 - cos2x) + \dfrac{5}{2}(1 + cos2x)$

$\rm \: f(x) = 11 - \dfrac{3}{2} + \dfrac{3}{2} cos2x + \dfrac{5}{2} + \dfrac{5}{2} cos2x$

$\rm \: f(x) = 11+ \bigg(\dfrac{3 + 5}{2}\bigg) cos2x + \dfrac{5 - 3}{2}$

$\rm \: f(x) = 11+ \bigg(\dfrac{8}{2}\bigg) cos2x + \dfrac{2}{2}$

$\rm \: f(x) = 11+ 4cos2x + 1$

$\rm \: f(x) = 4cos2x + 12$

Now, we know that,

$\rm \: - 1 \leqslant cos2x \leqslant 1$

On multiply by 4, each term we get

$\rm \: - 4 \leqslant 4cos2x \leqslant 4$

On adding 12 in each term, we get

$\rm \: 12 - 4 \leqslant 4cos2x + 12 \leqslant 4 + 12$

$\rm \: 8 \leqslant f(x) \leqslant 16$

$\rm\implies \:8 \leqslant 11 - {3sin}^{2}x + {5sin}^{2}x \leqslant 16$

So,

Minimum value of f(x) = 8

Maximum value of f(x) = 16

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ADDITIONAL INFORMATION

1. The maximum and minimum value of the expression

$\rm \: f(x) = a \: sinx \: + \: b \: cosx \: is \\ \\ \rm \: [ - \sqrt{ {a}^{2} + {b}^{2}}, \: \sqrt{ {a}^{2} + {b}^{2}}]$

$\rm \: f(x) = a \: sinx \: + \: b \: cosx \: + \: c \: is \\ \\ \rm \: [c - \sqrt{ {a}^{2} + {b}^{2}}, \: c + \sqrt{ {a}^{2} + {b}^{2}}]$

2. Range of Trigonometric functions

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

Answer 2

$\sf\underline {Question:-}$

$\sf \: find \: the \: maximum \: and \: minimum \\ \sf \: value : \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\sf\red{f(x) = 11 - 3 {sin}^{2}x + 5 {cos}^{2}x}$

$\sf\underline {Answer:-}$

$\sf \rightarrow \: please \: see \: the \: attatched \: file \:$