On what intervals is the function f(x)=x^7−7x^6 both decreasing and concave up?
To determine the intervals where the function f(x) = x^7 - 7x^6 is both decreasing and concave up, we need to check the sign of its first derivative and second derivative.
1. First, let's find the first derivative f'(x) of the given function:
f(x) = x^7 - 7x^6
Using the power rule of differentiation, we can find the derivative:
f'(x) = 7x^6 - 42x^5
2. Next, let's find the second derivative f''(x) of the function:
f'(x) = 7x^6 - 42x^5
Applying the power rule again, we can find the second derivative:
f''(x) = 42x^5 - 210x^4
3. To find the intervals where the function is decreasing, we need to find the intervals where the first derivative is negative (f'(x) < 0).
Set f'(x) < 0 and solve for x:
7x^6 - 42x^5 < 0
Factoring out 7x^5 from both terms:
7x^5(x - 6) < 0
Now, we have two critical points: x = 0 and x = 6.
Create a sign chart with the intervals (-∞, 0), (0, 6), and (6, +∞). Test a value from each interval in the inequality (x = -1, x = 1, x = 7):
For (-∞, 0): (-1)^5(-1 - 6) = -7 < 0 (True)
For (0, 6): (1)^5(1 - 6) = -5 > 0 (False)
For (6, +∞): (7)^5(7 - 6) = 343 > 0 (True)
Therefore, the intervals where f(x) = x^7 - 7x^6 is decreasing are (-∞, 0) and (6, +∞).
4. To find the intervals where the function is concave up, we need to find the intervals where the second derivative is positive (f''(x) > 0).
Set f''(x) > 0 and solve for x:
42x^5 - 210x^4 > 0
Factoring out 42x^4 from both terms:
42x^4(x - 5) > 0
Now, we have two critical points: x = 0 and x = 5.
Create a sign chart with the intervals (-∞, 0), (0, 5), and (5, +∞). Test a value from each interval in the inequality (x = -1, x = 1, x = 6):
For (-∞, 0): (42)(-1)^4(-1 - 5) = -252 < 0 (False)
For (0, 5): (42)(1)^4(1 - 5) = -168 < 0 (False)
For (5, +∞): (42)(6)^4(6 - 5) = 90 > 0 (True)
Therefore, the interval where f(x) = x^7 - 7x^6 is concave up is (5, +∞).
In conclusion, the function f(x) = x^7 - 7x^6 is both decreasing and concave up on the intervals (-∞, 0) and (6, +∞).
To determine the intervals on which the function f(x) = x^7 - 7x^6 is both decreasing and concave up, you need to find its derivative and second derivative.
Step 1: Find the derivative of f(x):
f'(x) = d/dx (x^7 - 7x^6)
To find the derivative, you apply the power rule:
f'(x) = 7x^6 - 42x^5
Step 2: Find the second derivative of f(x):
f''(x) = d/dx (7x^6 - 42x^5)
Again, applying the power rule:
f''(x) = 42x^5 - 210x^4
Now that we have the first and second derivatives, we can analyze them to determine the intervals on which f(x) is both decreasing and concave up.
Decreasing Interval:
To find where f(x) is decreasing, we set the derivative f'(x) = 0 and solve for x:
7x^6 - 42x^5 = 0
We can factor out the common term x^5:
x^5(7x - 42) = 0
This equation has two parts: x^5 = 0 and 7x - 42 = 0.
Solving x^5 = 0 gives us x = 0.
Solving 7x - 42 = 0, we find x = 6.
Therefore, f(x) is decreasing on the intervals (-∞, 0) and (6, ∞).
Concave Up Interval:
To find where f(x) is concave up, we set the second derivative f''(x) = 0 and solve for x:
42x^5 - 210x^4 = 0
We can factor out the common term x^4:
x^4(42x - 210) = 0
This equation has two parts: x^4 = 0 and 42x - 210 = 0.
Solving x^4 = 0 gives us x = 0.
Solving 42x - 210 = 0, we find x = 5.
Therefore, f(x) is concave up on the intervals (-∞, 0) and (5, ∞).
Combining the information, we can conclude that f(x) = x^7 - 7x^6 is both decreasing and concave up on the intervals (-∞, 0) and (6, ∞).