Find the open intervals on which f is increasing (decreasing).

Question

Let f(x)=4x^3+5. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates  of all relative maxima (minima)

1. f is increasing on which intervals?

2. f is decreasing on which intervals?

3. the relative maxima of f occurs at x=?

4. the relative minima of f occurs at x=?

Answer 1

When we're interested in increasing/decreasing, we have to take a look at the first derivative.

 

f(x) = 4x^3 + 5  --->  f'(x) = 12x^2

 

By setting f'(x) = 0 and solving for x, we can search for a relative min/max.  

 

f'(x) = 12x^2 = 0 ---> x = 0, so x is either a relative min or max

 

Next, pick a point less than 0 and greater than 0 to see the value of f'(x).  If it switches from negative to positive, it's a minimum.  If it switches from positive to negative, it's a maximum.  If it does neither, then it's not a maximum or minimum.  

 

Also, by checking values on either side of x = 0, you will determine if it is increasing or decreasing.  For example, f'(2) = 12*2^2 = 48 > 0, so it is increasing.

 

I hope this can get you started!