for x (-12, 10) the function f is defined:
f(x)=x^7(x+2)^2
On which two intervals is the function increasing (enter intervals in ascending order)?
Find the region in which the function is positive
Where does the function achieve its minimum?
it's turning points are when
[1] x*7 = 0 or x = 0
[2] (x+2)^2 = 0 or x - -2
The graph is increasing when
(-12 to -2) and (0 to 10)
To work this out find the turning points by finding f'[x]=0
ok hope that helps
To determine the intervals on which a function is increasing, we need to find the intervals where the derivative of the function, f'(x), is positive.
First, let's find the derivative of f(x):
f'(x) = 7x^6(x+2)^2 + 2x^7(2)(x+2)
Simplifying this expression, we get:
f'(x) = 7x^6(x+2)^2 + 4x^7(x+2)
Next, we set f'(x) > 0 to find the intervals where the function is increasing:
7x^6(x+2)^2 + 4x^7(x+2) > 0
To solve this inequality, we can factor out common terms:
x^6(x+2)[7(x+2) + 4x] > 0
x^6(x+2)[7x+14 + 4x] > 0
x^6(x+2)(11x + 14) > 0
Now, we can apply the sign chart method to find the intervals where the function is increasing:
On the number line, we mark points -2, -14/11, and 0:
- -2 -14/11 0 +
We can divide the number line into four intervals: (-∞, -2), (-2, -14/11), (-14/11, 0), and (0, ∞).
We then pick test values from each interval to check the sign of the expression x^6(x+2)(11x + 14).
For the interval (-∞, -2), we can pick x = -3:
(-3)^6(-3 + 2)(11(-3) + 14) > 0
729(-1)(-19) > 0
13851 > 0 (positive)
For the interval (-2, -14/11), we can pick x = -1:
(-1)^6(-1 + 2)(11(-1) + 14) > 0
1(1)(3) > 0
3 > 0 (positive)
For the interval (-14/11, 0), we can pick x = -10/11:
(-10/11)^6(-10/11 + 2)(11(-10/11) + 14) > 0
78320/1771561(-12/11) > 0
-94240/1771561 > 0 (negative)
For the interval (0, ∞), we can pick x = 1:
1^6(1 + 2)(11(1) + 14) > 0
1(3)(25) > 0
75 > 0 (positive)
Based on the sign chart analysis, the function f(x) is increasing on the intervals: (-∞, -2) and (-14/11, 0).
To find the region in which the function is positive, we can examine the sign of the function itself.
Since the function f(x) = x^7(x+2)^2 is a product of terms, it will be positive when all factors are positive or when an even number of factors are negative.
Let's analyze the sign of each factor:
1. x^7: This factor changes sign when x = 0, so it is negative for x < 0 and positive for x > 0.
2. (x+2)^2: This factor is always positive for any real value of x.
Based on this analysis, we need to consider when either factor is negative.
For x < 0, both factors are negative, which means f(x) < 0.
For x > 0, both factors are positive, which means f(x) > 0.
Hence, the function f(x) is positive for x > 0.
To find the minimum of the function, we need to locate the critical points where f'(x) = 0 or f'(x) is undefined. However, in this case, f'(x) is a polynomial and is always defined on the given interval.
Therefore, to find the minimum, we need to check the endpoints of the interval (-12, 10), which are x = -12 and x = 10, as well as any other possible local minimum or maximum points.
Let's evaluate the function at these points:
For x = -12:
f(-12) = (-12)^7(-12+2)^2 = 2985984
For x = 10:
f(10) = (10)^7(10+2)^2 = 3316137600
From these calculations, the function achieves its minimum value at x = -12 with a value of 2985984.
To summarize:
- The function f(x) is increasing on the intervals (-∞, -2) and (-14/11, 0).
- The function f(x) is positive for x > 0.
- The function achieves its minimum value of 2985984 at x = -12.