Question

Find the intervals in which function f(x) = `5x^(3/2) - 3x^(5/2)` is 

1. increasing
2. decreasing

Answer 1

f(x) = `5x^(3/2) - 3x^(5/2)`

f'(x) = `5 xx 3/2x^(3/2 - 1) - 3 xx 5/2 x^(5/2 - 1)`

f'(x) = `15/2x^((3 -2)/2) - 15/2 x^((5 - 2)/2)`

f'(x) = `15/2x^(1/2) - 15/2 x^(3/2)`

f'(x) = `15/2 sqrtx(1 - x)`

Now, f'(x) = 0

`15/2 sqrtx(1 - x) = 0`

`sqrtx = 0 or 1 - x = 0`

x = 0 or x = 1

(i) For increasing f'(x) > 0

`15/2 sqrtx(1 - x) > 0`

`sqrtx(1 - x) < 0`

f(x) is increasing, if x ∈ (0, 1)

(ii) For decreasing f'(x) < 0

`15/2 sqrtx(1 - x) < 0`

`sqrtx(1 - x) > 0`

f(x) is decreasing, if x ∈ (–∞, 0) or x ∈ (1, ∞) but x < 0 then function is not defined.

Hence, f(x) is decreasing only for x ∈ (1, ∞).