Find the critical points of

(4x-3)/(3x^(1/2))

you should say y = (4x-3)/(3x^(1/2))

or
f(x) = (4x-3)/(3x^(1/2))

dy/dx =[ 3x^(1/2) (4) - (4x-3)(3/2)x^(-1/2) ]/(9x)
= 0 for a max/min

12x^(1/2) = (3/2)(4x-3)/x^(1/2)
12x = (3/2)(4x - 3)
24x = 12x - 9
12x=-9
x = -3/4 , but x ≥0 or else x^(1/2) is undefined.

There are no turning points. (no max/min)

dy/dx can be reduced to (4x-3)/(6x^(3/2))
I will leave it up to you to show that the second derivative is
(9-4x)/(12x^(5/2) )

when we set that equal to zero , we get
x = 9/4
So there is a point of inflection when x = 9/4

what about intercepts?
let x = 0, y is undefined
let y = 0 , x = 3/4

I let Wolfram graph it, and got
http://www.wolframalpha.com/input/?i=y+%3D+(4x-3)%2F(3x%5E(1%2F2))+,+from+0+to+3

I am unable to explain why no point of inflection is shown, but my algebra shows that there is one.

it's because y"=0 when x = -9/4

but y is undefined for x<0.

check for +/- sign mistake

To find the critical points of the given function f(x) = (4x - 3)/(3√x), we need to determine the values of x where the derivative of the function is either zero or undefined.

Step 1: Find the derivative of the function f(x).
To find the derivative of f(x), we can use the quotient rule.
Using the quotient rule, the derivative of f(x) is given by:
f'(x) = [(3√x)(4) - (4x - 3)(3/2)x^(-1/2)]/(3√x)^2

Simplifying f'(x):
f'(x) = (12√x - 6x^(-1/2))/(9x)

Step 2: Determine where f'(x) is either zero or undefined.
Since the denominator can never be zero, we need to focus on the numerator: 12√x - 6x^(-1/2).

Setting the numerator equal to zero:
12√x - 6x^(-1/2) = 0

Multiplying through by x^(1/2):
12x - 6 = 0

Solving for x:
12x = 6
x = 1/2

Step 3: Determine if x = 1/2 is a critical point or an endpoint of the domain.
To determine if x = 1/2 is a critical point or an endpoint, we need to check the domain of the original function.
Since the original function f(x) is defined for all positive values of x, the critical point x = 1/2 is within the domain.

Therefore, the critical point of the function f(x) = (4x - 3)/(3√x) is x = 1/2.