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If the area bounded by f(x)=sqrt(tan x),...

If the area bounded by `f(x)=sqrt(tan x), y=f(c), x=0 and x=a, 0ltcltalt(pi)/(2)` is minimum then find the value of c.

Text Solution

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Graph of `f(x)=sqrt(tan x)` is as shown in the following figure.

Required area is
`A=overset(c)underset(0)int(f(c)-f(x))dx+overset(a)underset(c)int(f(x)-f(c))dx`
`=overset(c)underset(0)intsqrt(tan c)dx-overset(c)underset(0)intsqrt(tan x )dx +overset(a)underset(c)intsqrt(tan x)dx-overset(a)underset(c)intsqrt(tan c)dx`
`=(2c-a)sqrt(tan c)-overset(c)underset(0)intsqrt(tan x)dx +overset(a)underset(c)intsqrt(tan x)dx`
`therefore" "(dA)/(dc)=2sqrt(tan c)+(2c-a)(sec^(2)c)/(2sqrt(tan c))-sqrt(tan c)-sqrt(tan c)`
`=(2c-a)(sec^(2)c)/(2sqrt(tan c))`
`(dA)/(dc)=0rArrc=(a)/(2)`
`"At "c=(a)/(2), (dA)/(dc)` changes sign from negative to positive
Hence A is minimum when `c=(a)/(2)`
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