Question

Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).

Answer 1

f '(x) = `sqrt"x"`     ....[Given]

f(x) = ∫ f '(x)

`= int sqrt"x"` dx

`= int "x"^(1/2)` dx

`= "x"^(3/2)/(3/2)` + c

∴ f(x) = `2/3 "x"^(3/2) + "c"` ...(i)

Now, f(1) = 2      ....[Given]

∴ `2/3 (1)^(3/2) + "c" = 2`

∴ c = `2 - 2/3`

∴ c = `4/3`

∴ f(x) = `2/3 "x"^(3/2) + 4/3`