Consider the function:
f(x) = x/(x^2 + 2)
(a) Determine the intervals on which the function is concave up. Enter the leftmost interval first (e.g. (1,2) would come before (3,4)).
(b) Determine the intervals on which the function is concave down. Enter the leftmost interval first.
i only got to figure out the first derivative. And i did figure out second derivative but it came out to be some thing very long and nasty. so please help!
f'(x)=((x^2+2)-2x)/(x^2 + 2)^2
always simplify your first derivative before trying to take the second deriv.
I had it as
f' (x) = (2-x^2)/(x^2 + 2)^2
f'' (x) =[ (x^2+2)^2(-2x) - (2-x^2)(2)(x^2=2)(2x)]/ (x^2+2)^4
this factored to
-2x(x^2+2)[x^2+2 + 2(2-x^2)]/(x^2+2)^4
= -2x(6-x^2)/(x^2+2)^3
now the denominator cannot be zero and will always be positive, so as far as signs are concerned, we could ignore it.
so the curve is concave up when
-2x(6-x^2) > 0 or
2x(6-x^2) < 0
and the curve is concave down when
-2x(6-x^2) < 0 or
2x(6-x^2) > 0
can you take it from there?
btw, the points of inflection happen when
x = 0 or x = ±√6
To determine the intervals on which the function is concave up or concave down, you need to analyze the second derivative of the function.
Let's start by finding the second derivative of the given function, f(x).
First, we need to differentiate the function f(x) to find the first derivative, f'(x), which you already calculated correctly:
f'(x) = ((x^2 + 2) - 2x)/(x^2 + 2)^2
Now, to find the second derivative, we need to differentiate f'(x) again. Let's calculate it step by step:
1. Expand the numerator:
f'(x) = (x^2 + 2 - 2x)/(x^2 + 2)^2
= (x^2 - 2x + 2)/(x^2 + 2)^2
2. Apply the quotient rule:
(f'(x))' = [(x^2 + 2)^2 * (2x - 2) - (x^2 - 2x + 2) * (2(x^2 + 2)(2x))]/((x^2 + 2)^2)^2
To simplify this expression, we can observe that the denominator, (x^2 + 2)^2, is squared. Therefore, we can rewrite the denominator as (x^2 + 2)^4. Doing so, we have:
(f'(x))' = [(x^2 + 2)^2 * (2x - 2) - (x^2 - 2x + 2) * (2(x^2 + 2)(2x))]/(x^2 + 2)^4
Simplifying the numerator further, we get:
(f'(x))' = [(2x^3 + 4x - 2x - 4) - (4x^3 + 8x^2 - 8x + 4) * (2x)]/(x^2 + 2)^4
= (-8x^3 - 16x^2 + 16x - 4 - 8x^4 - 16x^3 + 16x^2 - 8x)/(x^2 + 2)^4
= (-8x^4 - 24x^3 + 16x - 4)/(x^2 + 2)^4
Now that we have the second derivative, (f'(x))', we can determine the intervals of concavity.
To find the intervals of concavity:
(a) Look for the points where the second derivative is equal to zero or undefined.
(b) Determine the intervals between these points where the second derivative has the same sign.
(c) If the second derivative is positive (greater than zero) within an interval, the function is concave up in that interval. If the second derivative is negative (less than zero) within an interval, the function is concave down in that interval.
Now, to analyze the second derivative function, (-8x^4 - 24x^3 + 16x - 4)/(x^2 + 2)^4, you can proceed with two methods:
1. Analyzing the signs:
To find the intervals of concavity, you can create a number line and determine the sign of the second derivative in different intervals.
Alternatively, you can use technology or a graphing calculator to plot the second derivative function and visually identify the intervals of concavity.
2. Factoring the second derivative:
Another approach is to factor the second derivative function and find the critical points.
You can attempt to factor the second derivative, (-8x^4 - 24x^3 + 16x - 4). If it is factorable, you can find the critical points and determine the intervals of concavity accordingly. However, in some cases, factoring may not be possible or practical, leading to complex roots.
Regardless of the method you use, analyzing the signs of the second derivative or plotting it with technology should help you determine the intervals of concavity and answer the question.
Remember, concave up intervals are where the second derivative is positive, and concave down intervals are where the second derivative is negative.