Consider the function f(x)=(x^2)e^(14x)
f(x) has two inflection values at x = C and x = D with C≤D
where C is
and D is
Finally for each of the following intervals, tell whether f(x) is concave up or concave down.
(−∞,C]:
[C,D]:
[D,∞)
f(x)=
x3
3−
x2
2−2x+7
f(x)=x33−x22−2x+7
.
To find the inflection points of a function, we need to determine where the concavity changes.
First, we find the second derivative of f(x) to determine concavity.
f'(x) = d/dx[(x^2)e^(14x)]
= (2x)e^(14x) + (x^2)(14e^(14x))
f''(x) = d/dx[(2x)e^(14x) + (x^2)(14e^(14x))]
= (2)e^(14x) + (2x)(14e^(14x)) + (2x)(14e^(14x)) + (x^2)(14^2e^(14x))
= 2e^(14x) + 28xe^(14x) + 196x^2e^(14x)
= e^(14x)(196x^2 + 28x + 2)
To find the inflection points, we need to solve f''(x) = 0.
e^(14x)(196x^2 + 28x + 2) = 0
Since e^(14x) is always positive for any value of x, we can ignore it in the equation.
196x^2 + 28x + 2 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 196, b = 28, and c = 2.
x = (-(28) ± √((28)^2 - 4(196)(2))) / 2(196)
= (-28 ± √(784 - 1568)) / 392
= (-28 ± √(-784)) / 392
= (-28 ± 28i) / 392
= -1/14 ± i/14
Since the solution is complex (involving the imaginary unit), there are no real inflection points. In other words, there are no values of C and D that satisfy C≤D.
For the intervals (−∞,C], [C,D], and [D,∞), we need to determine the concavity of f(x).
We can choose a test value within each interval and substitute it into f''(x) to determine the sign.
For the interval (−∞,C]:
Choose a value x1 < C and substitute it into f''(x)
If f''(x1) > 0, then f(x) is concave up on (−∞,C]
If f''(x1) < 0, then f(x) is concave down on (−∞,C]
For the interval [C,D]:
Choose a value x2 between C and D and substitute it into f''(x)
If f''(x2) > 0, then f(x) is concave up on [C,D]
If f''(x2) < 0, then f(x) is concave down on [C,D]
For the interval [D,∞):
Choose a value x3 > D and substitute it into f''(x)
If f''(x3) > 0, then f(x) is concave up on [D,∞)
If f''(x3) < 0, then f(x) is concave down on [D,∞)
Since we haven't found the values of C and D, we cannot determine the concavity of the function on the given intervals.