7) Consider the function f(x)=x2e4x.
For this function there are three important intervals: (−∞,A], [A,B], and [B,∞) where A and B are the critical numbers.
Find A
and B
For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A]:
[A,B]:
[B,∞)
since e^4x is always positive, and
f'(x) = 2x e^4x (2x+1)
f'=0 when x=0 or x = -1/2
f' < 0 where x < -1/2
f' > 0 where -1/2 < x < 0
f' > 0 where x > 0
This is clearly shown at
http://www.wolframalpha.com/input/?i=+x^2+e^%284x%29+for+-1+%3C+x+%3C+0.3
A: I don't know, A just never showed up for our coffee date.
B: Oh, B? He's hiding in the alphabet soup, trying to avoid his responsibilities.
For the intervals:
(−∞,A]: They're moving in sync, like a synchronized swimming team. They're both decreasing (DEC).
[A,B]: They've sorted out their differences and are now increasing (INC) together.
[B,∞): They're on the rise, feeling confident and increasing (INC) the fun.
To find the critical numbers A and B, we need to determine where the derivative of the function is equal to zero or does not exist.
Let's find the derivative of the function f(x) = x^2e^(4x) using the product rule and chain rule:
f'(x) = (2x)(e^(4x)) + (x^2)(4e^(4x))
= 2xe^(4x) + 4x^2e^(4x)
To find A, we need to find the value of x where f'(x) = 0 or does not exist.
Setting f'(x) = 0:
2xe^(4x) + 4x^2e^(4x) = 0
Factoring out e^(4x) and x:
x(2 + 4x)e^(4x) = 0
This equation will be satisfied if x = 0 or 2 + 4x = 0. Solving the second equation:
2 + 4x = 0
4x = -2
x = -1/2
So A = -1/2.
Now, let's analyze the sign of f'(x) in each interval.
For the interval (-∞, A], let's pick a value less than -1/2, for example, x = -2:
f'(-2) = 2(-2)e^(4(-2)) + 4(-2)^2e^(4(-2))
= -4e^(-8) + 16e^(-8)
= -4e^(-8) + 16e^(-8)
= 12e^(-8)
Since e^(-8) is positive, f'(-2) > 0.
Thus, f(x) is increasing in the interval (-∞, A].
For the interval [A, B], let's pick a value between -1/2 and the critical number B (which we will find later), for example, x = 0:
f'(0) = 2(0)e^(4(0)) + 4(0)^2e^(4(0))
= 0
Since the derivative is zero, f'(x) does not change sign in this interval, which means f(x) is neither increasing nor decreasing.
To find B, we can consider the second critical point where 2 + 4x = 0:
2 + 4x = 0
4x = -2
x = -1/2
So B = -1/2.
For the interval [B, ∞), let's pick a value greater than -1/2, for example, x = 1:
f'(1) = 2(1)e^(4(1)) + 4(1)^2e^(4(1))
= 2e^4 + 4e^4
= 6e^4
Since e^4 is positive, f'(1) > 0.
Thus, f(x) is increasing in the interval [B, ∞).
In summary:
A = -1/2
B = -1/2
(−∞,A]: INC
[A,B]: Neither INC nor DEC
[B,∞): INC
To find the critical numbers A and B for the function f(x) = x^2e^4x, we need to find the values of x where the derivative of the function equals zero or does not exist.
Step 1: Find the derivative of f(x)
f'(x) = (2xe^4x) + (x^2 * 4e^4x) = 2xe^4x + 4x^2e^4x
Step 2: Set f'(x) = 0 and solve for x
2xe^4x + 4x^2e^4x = 0
2x(e^4x + 2x*e^4x) = 0
Since 2x cannot be zero (as it would make the entire expression zero), we can divide both sides by 2x:
e^4x + 2x*e^4x = 0
Now we can factor out e^4x:
e^4x(1 + 2x) = 0
Setting each factor equal to zero gives us:
e^4x = 0 => There are no real solutions
1 + 2x = 0 => 2x = -1 => x = -1/2
So the critical number A is -1/2.
To find B, we need to check the behavior of the derivative as x approaches positive infinity. Since the derivative is a sum of terms involving exponential functions with positive exponents, as x goes to positive infinity, e^4x will dominate the behavior, and the derivative will approach positive infinity. Therefore, there is no critical number B for this function.
Now let's determine whether f(x) is increasing or decreasing on each of the given intervals:
1. (−∞, A]:
To determine if f(x) is increasing or decreasing on this interval, we need to choose a test point within the interval and evaluate the sign of the derivative. Let's choose x = 0, which is within the interval.
Substituting x = 0 into f'(x):
f'(0) = 2(0)e^(4(0)) + 4(0)^2e^(4(0)) = 0 + 0 = 0
Since f'(0) = 0, we conclude that f(x) is neither increasing nor decreasing on the interval (-∞, A].
2. [A, B]:
Since we found A = -1/2 and there is no critical number B, this interval does not exist.
3. [B, ∞)
Since there is no critical number B, this interval does not exist either.
In summary:
A = -1/2
[B, ∞) does not exist
[A, B] does not exist
(-∞, A] is neither increasing nor decreasing