Apply rolle’s theorem to f(x) = sinx(cos2x)^1/2 in the interval [o,pie/4]
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Answer:
f(x)=e
x
(sinx−cosx),xϵ[
4
π
,
4
5π
]
Sine, cosine and exponential function are always continuous.
∴ Given function is continuous in [
4
π
,
4
5π
]
Differentiating w.r. to x, we get
f
′
(x)=e
x
(cosx+sinx)+(sinx−cosx)e
x
=e
x
[cosx+sinx+sinx−cosx]
=2e
x
sinx
Which exists for all x.
f(π/4)=e
π/4
(
2
1
−
2
1
)=0
and f(5π/4)=e
5π/4
(−
2
1
+
2
1
)=0
∴f(π/4=f(5π/4)=0
∴ The given function statisfies all three condition of Rolle's theorem.
For maxima or minima
f
′
(x)=0
2e
x
sinx=0
sinx=0
x=nπ+(−1)
n
(0)
x=nπ
x=π
∵π lies between [
4
π
,
4
5π
] so Rolle's theorem is verified.
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