Use the given derivative to find all critical points of 'f' and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume that 'f' is continuous everywhere.
f' (x) = (1-2x)/ ∛(x+3)
You have given us the derivative, and it is only zero when x = 1/2.
The derivatve becomes more negative when x>1/2, so x=1/2 is a relative maximum of f(x).
To find the critical points of the function f and determine whether they correspond to a relative maximum, relative minimum, or neither, we need to follow these steps:
Step 1: Find the x-values where the derivative f'(x) is equal to zero or undefined. These are the potential critical points.
Step 2: Evaluate the sign of the derivative on each side of the critical points.
Step 3: Determine whether the function f has a relative maximum, relative minimum, or neither at each critical point based on the sign changes of the derivative.
Now let's proceed with the calculations.
Step 1: Find the potential critical points:
Setting the derivative f'(x) to zero:
(1 - 2x)/ ∛(x+3) = 0
To find the potential critical point, we need to solve the numerator (1 - 2x) equal to zero:
1 - 2x = 0
-2x = -1
x = 1/2
Step 2: Evaluate the sign of the derivative on each side of the critical point.
We consider three intervals: (-∞, 1/2), (1/2, ∞), and around any potential points where the derivative is undefined.
For x < 1/2, plug in a value smaller than 1/2 into f'(x):
If x = 0, f'(0) = (1 - 2(0))/ ∛(0+3) = 1/∛3 > 0
For x > 1/2, plug in a value larger than 1/2 into f'(x):
If x = 1, f'(1) = (1 - 2(1))/ ∛(1+3) = -1/2∛4 < 0
Step 3: Determine whether the function has a relative maximum, relative minimum, or neither at each critical point.
At x = 1/2, the derivative is zero, and we have a sign change.
Thus, x = 1/2 is a critical point of f.
When the derivative changes sign from positive to negative (as in this case), the function has a relative maximum at that point.
Therefore, the critical point x = 1/2 corresponds to a relative maximum of the function f.
To find the critical points of a function, we need to find the values of x where the derivative is either zero or undefined.
In this case, let's first find where the derivative is undefined. The derivative is undefined when the denominator of the fraction is zero.
∛(x+3) = 0
To find the values of x that satisfy this equation, we can cube both sides:
x + 3 = 0
Solving for x, we find:
x = -3
So the function is undefined at x = -3.
Now, let's find where the derivative is zero. The derivative is zero when the numerator of the fraction is zero:
1 - 2x = 0
Solving for x, we get:
2x = 1
x = 1/2
So the derivative is zero at x = 1/2.
Now that we have found the critical points (x = -3 and x = 1/2), we need to determine whether they correspond to a relative maximum, relative minimum, or neither.
To do this, we can analyze the sign of the derivative on either side of each critical point.
For x < -3, the denominator ∛(x+3) is negative, so the sign of the derivative depends on the sign of the numerator (1 - 2x). If x < 1/2, then 1 - 2x is positive, and thus the derivative is positive. If x > 1/2, then 1 - 2x is negative, and thus the derivative is negative.
For x > -3, the denominator ∛(x+3) is positive. Again, we look at the sign of the numerator (1 - 2x). If x < 1/2, then 1 - 2x is positive, and the derivative is negative. If x > 1/2, then 1 - 2x is negative, and the derivative is positive.
Now let's summarize our findings:
- At x = -3, the derivative is undefined. Therefore, it is neither a relative maximum nor a relative minimum.
- At x = 1/2, the derivative is zero. To determine whether it is a relative maximum or minimum, we need to perform a further test, such as the second derivative test.
By evaluating the signs of the derivative on either side of the critical point, we can conclude whether a relative maximum, relative minimum, or neither occurs.