increasing decreasinf function with derivatives

[craziebbygirl]

Hi I need help with this problem:

f(x)= 2X + 1 / X^(1/2)

Use first derivative to show the intervals on which the function is increasing or decreasing and find the extrema of the function

Thankyou!

I dount under stand how to find f' and f" x..

 

[mmm4444bot]

… f(x) = 2X + 1 / X^(1/2)

As you typed it, the definition for function f is the fraction 1/x^(1/2) plus the term 2x.

Is this correct?

I want to make sure that it's not supposed to be the following, instead.

f(x) = (2x + 1)/x^(1/2)

In this algebraic fraction, the numerator is 2x + 1.

Please confirm.

 

[craziebbygirl]

yes you are correct, also I think that F'(x)= 2 + 1/2X^-3/2.

Am I correct? If so I do not understand how to make that = 0 to find the critical values of the function.

 

[mmm4444bot]

… I think that F'(x)= 2 + 1/2X^-3/2.

Please type parentheses around fractional exponents.

x^-3/2 is not clear because it actually means (1/2) x^(-3).

Type x^(-3/2) instead.

I got a slightly different result for the derivative of f.

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

I used the Quotient Rule.

[D(2x + 1) x^(1/2) - (2x + 1) D{x^(1/2)}] / [x^(1/2)]^2

The critical values are those values of x that make f '(x) either zero or undefined. Extrema for f occur at those x values that make the derivative zero.

Function f is increasing on those intervals where f '(x) is positive, and it's decreasing on the intervals where f '(x) is negative. (Remember, the derivative is the slope. Positive slope: increasing; negative slope: decreasing.)

So, once you get the critical value(s), they divide the Real number line into intervals. Pick a test value from each inveral and use it to evaluate f prime. If f prime is positive for the test value, then it's positive for every number in the interval, and thus f is increasing on that interval. If f prime is negative, then f is decreasing on that interval.

Perhaps, it would be easier for you to see how to solve f '(x) = 0 if the expression were to be written with radical signs, instead.

The expression x^(-1/2) - (1/2) x^(-3/2) is equivalent to the following.

f '(x) = 1/sqrt(x) - 1/[2 sqrt(x^3)]

Simplify the radical sqrt(x^3).

Factor out 1/sqrt(x).

Set each factor equal to zero, and solve for x.

If you would like more help, please show your work, so that I might determine where to continue helping.

 

[craziebbygirl]

I have gotten:

0 = (2) - (1/2x^(-2/3))
2 = (1/2) * (1/x^(3/2))
4 = (1 / (x^(3/2))
(x^(2/3)) = (1 / (x^(3/2))

x = (1/4)^(3/2)
x=.3969

 

[Deleted member 4993]

Now what??

Sketch the graph in your graphing calculator.

Watch the behaviour of the graph around x = 0.3965