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## Differential Calculus

### Course: Differential Calculus>Unit 5

Lesson 3: Intervals on which a function is increasing or decreasing

# Finding decreasing interval given the function

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.1 (EK)
Sal finds the intervals where the function f(x)=x⁶-3x⁵ is decreasing by analyzing the intervals where f' is positive or negative.

## Want to join the conversation?

• Why are the intervals open, not closed?
• I think the interval should include 0. Because the graph is continuous. Can you find an x<0 makes f(x)<=f(0) or an 2.5>x>0 makes f(x)>=f(0)? Of course there are no such points.
Since none, the graph is decreasing around point (0,0) regardless of f'(0)=0,which tells us the graph is "horizontal"
If you must say 0 shouldn't appear in the decreasing interval, you could say that the 'strictly decreasing interval' doesn't include 0.
• Couldn't you use sign analysis?
• Yes you can! The next videos include them!
• Why was x<0 included in the intervals for x that lead to a negative differential at ? Unless I missed something in the video, I didn't see any explanation for this.
• we are looking for intervals which f is decreasing.
it means we find intervals for f'(x) < 0

since our f'(x) = x^4*(6x-15)
for x<0 our f'(x) will always show negative value.

ex) for x = -1, f'(-1) = 1*(-6-15) = -21
• Is the interval gonna be written as ] 5/2, 0 [ U ] 0, -infinity[ ?
• isn't the instantaneous speed at x = 0 is 0 so it isn't decreasing (at the moment).
• When we are solving 6x^5-15x^4>0, why can't we divide both sides by 3x^4 since x^4>0 (in this case x cannot be 0)? So we can solve it more easily.
• When we are solving equations for variables (which is essentially what we are doing here), if we divide by 3x^4, we remove possible answers. If we divided, we may forget that x can't = 0, and just be left with x <(5/2). As a general rule of thumb, never divide by a variable.
• I understand the function is decreasing for x between 0 and 2.5. I don't understand why x is decreasing for x<0. X^4 will always be positive. Also, if I graph the equation, the function is actually increasing for all values of x<0. Can anyone explain this?
• If you start at 0 and go towards negative infinity, then yes, all the values are increasing. However, we are talking about increasing in terms of slope, so we move from left to right. If you started at negative infinity and moved towards 0, then all the values would be decreasing and there slope of the tangent line will be negative. I hope this helps!
• At , why couldn't you take the 4th root of both sides of x^4 < 0 and get x < 0?
• If you did this, you would have to use two inequalities for the +/- symbol, x<+0 and x>-0. This is the same thing as saying x<0 and x>0, or x=/=0.