f(r)=r/(10r^2+7) find the critical numbers. I found the derivative to be -10r^2+7/(10r^2+7)^2 and now I am stuck. What do I do next?
so, the critical numbers are the values where f'(r) = 0, right?
A fraction is zero if its numerator is zero.
So, what is r if
7 - 10r^2 = 0?
I made that seem a lot more complicated than it was! Thank you for your help!
To find the critical numbers of a function, we need to find the values of the independent variable (in this case, "r") where the derivative is equal to zero or undefined. Let's continue from where you left off:
The derivative you found is correct: f'(r) = (-10r^2 + 7) / (10r^2 + 7)^2
To further determine the critical numbers, we need to set the derivative equal to zero and solve for "r":
(-10r^2 + 7) / (10r^2 + 7)^2 = 0
Now, we can observe that the numerator (-10r^2 + 7) can be zero, but the denominator (10r^2 + 7)^2 cannot be zero. So, we set the numerator equal to zero:
-10r^2 + 7 = 0
To solve this equation, we isolate the variable "r":
-10r^2 = -7
r^2 = 7/10
To find the solutions, we take the square root of both sides (considering both positive and negative roots):
r = ±√(7/10)
Therefore, the critical numbers for the function f(r) = r/(10r^2 + 7) are ±√(7/10).
These values represent the points on the function where the slope changes, which are crucial in identifying maxima, minima, or any points of inflection.