Find the points of inflexion for f(x) =(sinx+cosx)
Discuss the concavity and convexity.
Answers
first derivative
on solving we get
.......(1)
Second derivative
........(2)
divide and multiplying by √2
=2√2
=2√2
1/√2=cosπ/4
1/√2=sinπ/4
from identity cos(A+B)=cosA cosB-sinA sinB
So applying this identity
2√2.....(3)
3rd derivative
or
These are the point of inflexion.
❤_________ 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