Question

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

Answer 1

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

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

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

1/√2=cosπ/4

1/√2=sinπ/4

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

So applying this identity

2√2$e^x$$cos(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.

Answer 2

f(x)=(sinx+cosx)e

x

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

These are the point of inflexion

hope it helps you