Math, asked by jangaramulu044, 4 months ago

find max and min value of function f(x)=7cosx-24sinx+5​

Answers

Answered by hrn21agmailcom
1

Answer:

12 & -19

Step-by-step explanation:

f(x) = 7 cos x - 24 sin x + 5

f '(x) = - 7 sin x -24 cos x

for f ' (x) = 0

- 7 sin x -24 cos x = 0

7 sinx = - 24 cos x

for f " (x) = - 7 cos x + 24 sin x

i,e. f " (x) = - 7 cos x + 24 ( - 24/7 cos x )

f " (x) = - 625/7 cos x

for x = 0° f " (x) = - 625/7 < 0 & f is max

for x = 90° f " (x) = 0 & f is max/min

for x = 180° f " (x) = 625/7 > 0 f is min

hence...

f(0°) = 7 cos 0° -24sin 0° + 5= 12 (max)

f(90°)= 7cos 90°-24sin90°+5 = -19 (min)

f(180°) =7cos 180° -24 sin180° +5 = -2(min)

therefore...

f max = 12 & f min = -19

Answered by Anonymous
11

 \bull \:  \:  \:  \rm \underline \bold{Given :}

 \sf \mapsto \: f(x) =  \: 7 \cos \: x  \: - 24 \:  \sin \: x \:  +  \: 5 \\

 \bull \:  \:  \:  \rm \underline \bold{Solution :}

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf  \: f(x) =  \: 7 \cos \: x  \: - 24 \:  \sin \: x \:  +  \: 5 \\   \\

 \sf \: \boxed{} \:  f(x)  \: will  \: be \:max \:   if \:  -24  \:  \sin \:  x  \: =  \: 0

 \sf hence,

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf \: f(x)_{max}\:= 7 \cos \: x - 24 \sqrt{1 -  { \cos}^{2}x } +  5

 \sf \: Putting   \:  value \:  of \:   \cos \:  x \:  \\  \sf \: so \: that \:  - 24 \sqrt{1 -  { \cos}^{2}x }  = 0

 \implies  \sf \: f(x) = 7  \times 1 - 24 \times 0 + 5

 \implies  \sf \: { \underline {\boxed {\sf{ f(x)_{max}= 12}}}}

Now,

 \sf \: \boxed{} \:  f(x)  \: will  \: be \:minimum \:   if \:  7 \:  \cos \:  x  \: =  \: 0

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf \: f(x)_{max}\:= 7 \cos \: x - 24 \sqrt{1 -  { \cos}^{2}x } +  5

 \sf hence,

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf \: f(x)_{max}\:= 7  \sqrt{1 -  { \sin}^{2}x } \:  - 24  \sin \: x+  5

 \sf \: Putting  \:  value \:  of \:   \cos \:  x \:  \\  \sf \: so \: that  \: woud\:  7 \sqrt{1 -  { \sin}^{2}x }  = 0

 \implies  \sf \: f(x)_{min}\:= 7  \sqrt{1 -  1 } \:  - 24 \times   1+  5 \\

 \implies \sf \: \: f(x)_{min}\: =  - 24 + 5

 \implies   { \underline{\boxed{\sf \: {f(x)_{min}\: =  - 19}}}}

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HOPE THIS IS HELPFUL...

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