Question

verify cauchy's mean value theorem for f(x)= sinx and g(x) = cosx in interval [a, b]​

Answer 1

Answer:

Given that f(x)=sinx,g(x)=cosx on [0,π/2]

Thus f(x) &g(x) is continuous on[0,π/2].

f(x)=sinx.

f'(x)=cosx

g(x)=cosx

g'(x)=-sinx

:. f(x) &g(x) are differentiable on [0,π/2].

Here g'(x) not equal to 0

Three conditions of cauchy's mean value theorém are verified.

Then their exists a point c€(a,b) , such that

f(b)-f(a)/g(b)-g(a)=f'(c)/g'(c)

f(π/2)-f(0)/g(π/2)-g(0)=f'(c)/g'(c)

sinπ/2-sin0÷cosπ/2-cos0=cosx/-sinc

1-0/0-1=cosc/-sinc

cosc/sinc=1

cosc=sinc

sin(π/2-c)=sinc

2c=π/2

c

C=π/4€(0,π/2).

:. Cauchy's mean value theorem is verified.