French, asked by 7TeeN, 10 months ago

Find the points of inflexion for f(x) =(sinx+cosx)
 f(x) = (sinx + cosx) {e}^{x} \: where \: \\ belongs \: to \: 0 \: to \: 2\pi \:
Discuss the concavity and convexity.​

Answers

Answered by MRsteveAustiN
5

 f(x) = (sinx + cosx) {e}^{x}

first derivative

 f'(x) = (-cosx+ sinx) {e}^{x}+= (sinx + cosx) {e}^{x}

on solving we get

 f(x) = 2cosx {e}^{x}.......(1)

Second derivative

 f''(x) = 2e^x (cosx -sinx)........(2)

divide and multiplying by √2

=2√2e^x (1/\sqrt{2}×cosx -1/\sqrt{2}×sinx)

=2√2e^x (cos\pi/4×cosx -sin\pi/4×sinx)

1/√2=cosπ/4

1/√2=sinπ/4

from identity cos(A+B)=cosA cosB-sinA sinB

So applying this identity

2√2e^xcos(x+\frac{\pi}{4}).....(3)

3rd derivative

f'''=2e^x (cosx -sinx)+2e^x(-sinx-cosx)

f'''=2e^x [(cosx -sinx-sinx-cosx)]

f'''=2(-2sinx)e^x

f'''=-4sinxe^x........(3)

 equating (2)=0

  2e^x (cosx -sinx)=0

   (cosx -sinx)=0

   tanx=1

   tanx=tan\pi/4

tan(\pi+\pi/4)

or  tan5\pi/4

x=\pi/4,5\pi/4

These are the point of inflexion.

Answered by Aɾꜱɦ
7

❤_________ ANSWER ________❤

first derivative

f'(x) = (-cosx+ sinx) {e}^{x}+= (sinx + cosx) {e}^{x}f

(x)=(−cosx+sinx)e

x

+=(sinx+cosx)e

x

on solving we get

f(x) = 2cosx {e}^{x}f(x)=2cosxe

x

.......(1)

Second derivative

f''(x) = 2e^x (cosx -sinx)f

′′

(x)=2e

x

(cosx−sinx) ........(2)

divide and multiplying by √2

=2√2e^x (1/\sqrt{2}×cosx -1/\sqrt{2}×sinx)e

x

(1/

2

×cosx−1/

2

×sinx)

=2√2e^x (cos\pi/4×cosx -sin\pi/4×sinx)e

x

(cosπ/4×cosx−sinπ/4×sinx)

1/√2=cosπ/4

1/√2=sinπ/4

from identity cos(A+B)=cosA cosB-sinA sinB

So applying this identity

2√2e^xe

x

cos(x+\frac{\pi}{4})cos(x+

4

π

) .....(3)

3rd derivative

f'''=2e^x (cosx -sinx)+2e^x(-sinx-cosx)f

′′′

=2e

x

(cosx−sinx)+2e

x

(−sinx−cosx)

f'''=2e^x [(cosx -sinx-sinx-cosx)]f

′′′

=2e

x

[(cosx−sinx−sinx−cosx)]

f'''=2(-2sinx)e^xf

′′′

=2(−2sinx)e

x

f'''=-4sinxe^x........(3)f

′′′

=−4sinxe

x

........(3)

equating (2)=0equating(2)=0

2e^x (cosx -sinx)=02e

x

(cosx−sinx)=0

(cosx -sinx)=0(cosx−sinx)=0

tanx=1tanx=1

tanx=tan\pi/4tanx=tanπ/4

tan(\pi+\pi/4)tan(π+π/4)

or tan5\pi/4tan5π/4

x=\pi/4,5\pi/4x=π/4,5π/4

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