Question

If p (x) =sinx(sin3x+3) +cosx(cos3x+4)+1/2 sin^2 2x+5 find the range of p (x)

Answer 1

SOLUTION

TO DETERMINE

The range of

$\displaystyle \sf{p(x) = \sin x( { \sin}^{3}x + 3) + \cos x( { \cos}^{3}x + 4) + \frac{1}{2} { \sin}^{2} 2x + 5}$

FORMULA TO BE IMPLEMENTED

$\sf{ - \sqrt{ {a}^{2} + {b}^{2} } \leqslant a \sin x + b \cos x \leqslant \sqrt{ {a}^{2} + {b}^{2} } }$

EVALUATION

$\displaystyle \sf{p(x) = \sin x( { \sin}^{3}x + 3) + \cos x( { \cos}^{3}x + 4) + \frac{1}{2} { \sin}^{2} 2x + 5}$

$\displaystyle \sf{ = ( { \sin}^{4}x + 3 \sin x) + ( { \cos}^{4}x + 4 \cos x) + \frac{1}{2} {(2 \sin x \cos x)}^{2} + 5}$

$\displaystyle \sf{ = ( { \sin}^{4}x + { \cos}^{4}x) + (3 \sin x + 4 \cos x) + 2 { \sin}^{2} x \: { \cos}^{2}x + 5}$

$\displaystyle \sf{ = ( { \sin}^{4}x + 2 { \sin}^{2} x \: { \cos}^{2}x + { \cos}^{4}x) + (3 \sin x + 4 \cos x) + 5}$

$\displaystyle \sf{ = {( { \sin}^{2} x \: + { \cos}^{2}x )}^{2} + (3 \sin x + 4 \cos x) + 5}$

$\displaystyle \sf{ = {( 1 )}^{2} + (3 \sin x + 4 \cos x) + 5}$

$\displaystyle \sf{ = (3 \sin x + 4 \cos x) + 6}$

Now

$\sf{ - \sqrt{ {3}^{2} + {4}^{2} } \leqslant 3 \sin x + 4 \cos x \leqslant \sqrt{ {3}^{2} + {4}^{2} } }$

$\sf{ \implies - \sqrt{ 25 } \leqslant 3 \sin x + 4 \cos x \leqslant \sqrt{25 } }$

$\sf{ \implies -5 \leqslant 3 \sin x + 4 \cos x \leqslant 5 }$

$\sf{ \implies -5 + 6 \leqslant 3 \sin x + 4 \cos x + 6\leqslant 5 + 6 }$

$\sf{ \implies 1\leqslant 3 \sin x + 4 \cos x + 6\leqslant 11 }$

$\sf{ \implies 1\leqslant p(x)\leqslant 11 }$

Hence the required Range of the function is

$\sf{ \{ \: p(x) \: : \: 1\leqslant p(x)\leqslant 11 \: \}}$

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