Verify Rolles theorem for function `f(x)=s in^4\ x+cos^4\ x` on `[0,\ pi/2]`

Answer

N/a

Explanation

We have, `f(x) = sin^(4)x+cos^(4) x` in `[0,(pi)/(2)]`
(i) `f(x)` is continuous in `[0,(pi)/(2)]`
[since, `sin^(4)x` and `cos^(4)x` are continuous functions and we know that, if g and h be continuous function)
(ii) `f'(x) = 4(sinx)^(3).cosx+4(cosx)^(3).(-sinx)`
`= 4sin^(3)x.cosx-4sinx.cos^(3)x`
`= 4sinxcosx(sin^(2)x-cos^(2)x)` which exists in `(0,(pi)/(2))`
Hence, `f(x)` is differentiable in `(0,(pi)/(2))`
(iii) Also, `f(0) = 0+1` and ` f'(pi/2)=1+0 = 1`
`rArr f(0) = f((pi)/(2))`
Conditions of Rolle's theorem are satisfied.
Hence, there exists atleast one `c in (0,(pi)/(2))` such that `f'(c) = 0`.
`:. 4 sinx cosc (sin^(2)c - cos^(2)c) = 0`
` rArr 4sinc cosc (-cos2c)=0`
`rArr -2sin2c.cos2c = 0`
`rArr -sin4c = 0`
`rArr sin4c = 0`
`rArr 4c = pi`
`rArr c = (pi)/(4)`
and `(pi)/(4) in (0,(pi)/(2))`
Hence, Rolle's theorem has been verified.