f(x)= 3+4xe^{-5x}complete the sentences concerning the function
a) the function f is decresing on the iterval......
b) the function f is increacing on the interval
c) the function f is concave down on the interval
d) the function f is concave up on the interva
f is decreasing when f' is negative
and similarly for increasing
f is concave down when f" is negative
and similarly for concave up
f = 3 + 4xe^-5x
f' = 4(1-5x)e^-5x
f" = 20(5x-2)e^-5x
Hint: e^-5x is never zero
To determine whether the function f(x) = 3 + 4xe^(-5x) is increasing or decreasing on a given interval, we need to analyze its derivates. We'll find the first derivative to determine whether it is increasing or decreasing and the second derivative to determine its concavity.
a) To find the interval on which the function f is decreasing, we need to find the intervals where the first derivative is negative. We can do this by taking the first derivative of f(x) with respect to x:
f'(x) = 4e^(-5x) - 20xe^(-5x)
Now, set f'(x) less than zero (f'(x) < 0) and solve for x to find the interval on which f is decreasing.
4e^(-5x) - 20xe^(-5x) < 0
Factoring out e^(-5x), we have:
e^(-5x)(4 - 20x) < 0
For this inequality to be true, either e^(-5x) and (4 - 20x) must have opposite signs or one must be zero. Since e^(-5x) is always positive, we need (4 - 20x) < 0.
4 - 20x < 0
Solving this inequality gives us:
x > 1/5
Therefore, the function f is decreasing on the interval (1/5, infinity).
b) To find the interval on which the function f is increasing, we can use the same approach but instead set f'(x) greater than zero (f'(x) > 0) and solve for x.
4e^(-5x) - 20xe^(-5x) > 0
e^(-5x)(4 - 20x) > 0
For this inequality to be true, both e^(-5x) and (4 - 20x) must have the same sign or one of them should be zero. Since e^(-5x) is always positive, we need (4 - 20x) > 0.
4 - 20x > 0
Solving this inequality gives us:
x < 1/5
Therefore, the function f is increasing on the interval (-infinity, 1/5).
c) To determine the interval on which the function f is concave down, we need to examine its second derivative. Taking the second derivative of f(x):
f''(x) = 20e^(-5x) - 100xe^(-5x) - 100e^(-5x)
Simplify this expression:
f''(x) = -100xe^(-5x) + 20e^(-5x) - 100e^(-5x)
Now, to find the interval on which f is concave down, we set f''(x) less than zero (f''(x) < 0) and solve for x.
-100xe^(-5x) + 20e^(-5x) - 100e^(-5x) < 0
Simplifying further:
-100xe^(-5x) - 80e^(-5x) < 0
Factor out e^(-5x):
e^(-5x)(-100x - 80) < 0
To satisfy the inequality, e^(-5x) and (-100x - 80) must have different signs. We focus on the expression (-100x - 80) and solve for x:
-100x - 80 < 0
Solving this inequality:
x > -4/5
Therefore, the function f is concave down on the interval (-4/5, infinity).
d) To determine the interval on which the function f is concave up, we can set the second derivative greater than zero (f''(x) > 0) and solve for x.
-100xe^(-5x) + 20e^(-5x) - 100e^(-5x) > 0
Simplifying:
-100xe^(-5x) - 80e^(-5x) > 0
Factor out e^(-5x):
e^(-5x)(-100x - 80) > 0
To satisfy the inequality, e^(-5x) and (-100x - 80) must have the same sign. We focus on the expression (-100x - 80) and solve for x:
-100x - 80 > 0
Solving this inequality:
x < -4/5
Therefore, the function f is concave up on the interval (-infinity, -4/5).
In summary:
a) The function f is decreasing on the interval (1/5, infinity).
b) The function f is increasing on the interval (-infinity, 1/5).
c) The function f is concave down on the interval (-4/5, infinity).
d) The function f is concave up on the interval (-infinity, -4/5).