Answer :
To solve the problem of finding the inflection points of the function [tex]\( f(x) = 12x^5 + 60x^4 - 240x^3 + 3 \)[/tex] and determining its concavity, follow these steps:
### Step 1: Find the Second Derivative
The concavity of a function is determined by its second derivative. First, calculate the second derivative of [tex]\( f(x) \)[/tex].
1. First derivative:
Start by differentiating the function [tex]\( f(x) \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(12x^5 + 60x^4 - 240x^3 + 3) = 60x^4 + 240x^3 - 720x^2
\][/tex]
2. Second derivative:
Differentiate [tex]\( f'(x) \)[/tex] to find the second derivative, [tex]\( f''(x) \)[/tex]:
[tex]\[
f''(x) = \frac{d}{dx}(60x^4 + 240x^3 - 720x^2) = 240x^3 + 720x^2 - 720x
\][/tex]
### Step 2: Find Inflection Points
Inflection points occur where [tex]\( f''(x) = 0 \)[/tex] or is undefined. Solve the equation [tex]\( f''(x) = 0 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
240x^3 + 720x^2 - 720x = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
240x(x^2 + 3x - 3) = 0
\][/tex]
This gives us:
[tex]\[
x = 0 \quad \text{or} \quad x^2 + 3x - 3 = 0
\][/tex]
Solve the quadratic equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -3 \)[/tex]. This results in:
[tex]\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{9 + 12}}{2}
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{21}}{2}
\][/tex]
The solutions are:
[tex]\[
x = \frac{-3 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{21}}{2}
\][/tex]
Numerically, these solutions are approximately [tex]\( x = 0.7913 \)[/tex] and [tex]\( x = -3.7913 \)[/tex].
The inflection points [tex]\( x = D, E, F \)[/tex] are therefore [tex]\( x = -3.7913, 0, 0.7913 \)[/tex].
### Step 3: Determine Concavity
Check the intervals defined by these inflection points to determine if the function is concave up or concave down in each interval:
1. Interval [tex]\((-∞, D) = (-∞, -3.7913)\)[/tex]:
Test a point like [tex]\( x = -1000 \)[/tex].
- If [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- Here, [tex]\( f''(-1000) < 0 \)[/tex], so it is concave down.
2. Interval [tex]\((D, E) = (-3.7913, 0)\)[/tex]:
Test a point between the inflection points, such as [tex]\( x = -2 \)[/tex].
- If [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
- Here, [tex]\( f''(-2) > 0 \)[/tex], so it is concave up.
3. Interval [tex]\((E, F) = (0, 0.7913)\)[/tex]:
Choose a point like [tex]\( x = 0.5 \)[/tex].
- If [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- Here, [tex]\( f''(0.5) < 0 \)[/tex], so it is concave down.
4. Interval [tex]\((F, ∞) = (0.7913, ∞)\)[/tex]:
Test a point in this interval, such as [tex]\( x = 1000 \)[/tex].
- If [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
- Here, [tex]\( f''(1000) > 0 \)[/tex], so it is concave up.
### Conclusion
Based on these results, the inflection points and concavity of the function are as follows:
- Inflection points:
[tex]\( D = -3.7913 \)[/tex], [tex]\( E = 0 \)[/tex], [tex]\( F = 0.7913 \)[/tex]
- Concavity:
- [tex]\((-∞, D)\)[/tex]: Concave down
- [tex]\((D, E)\)[/tex]: Concave up
- [tex]\((E, F)\)[/tex]: Concave down
- [tex]\((F, ∞)\)[/tex]: Concave up
### Step 1: Find the Second Derivative
The concavity of a function is determined by its second derivative. First, calculate the second derivative of [tex]\( f(x) \)[/tex].
1. First derivative:
Start by differentiating the function [tex]\( f(x) \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(12x^5 + 60x^4 - 240x^3 + 3) = 60x^4 + 240x^3 - 720x^2
\][/tex]
2. Second derivative:
Differentiate [tex]\( f'(x) \)[/tex] to find the second derivative, [tex]\( f''(x) \)[/tex]:
[tex]\[
f''(x) = \frac{d}{dx}(60x^4 + 240x^3 - 720x^2) = 240x^3 + 720x^2 - 720x
\][/tex]
### Step 2: Find Inflection Points
Inflection points occur where [tex]\( f''(x) = 0 \)[/tex] or is undefined. Solve the equation [tex]\( f''(x) = 0 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
240x^3 + 720x^2 - 720x = 0
\][/tex]
Factor out the greatest common factor:
[tex]\[
240x(x^2 + 3x - 3) = 0
\][/tex]
This gives us:
[tex]\[
x = 0 \quad \text{or} \quad x^2 + 3x - 3 = 0
\][/tex]
Solve the quadratic equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -3 \)[/tex]. This results in:
[tex]\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{9 + 12}}{2}
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{21}}{2}
\][/tex]
The solutions are:
[tex]\[
x = \frac{-3 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{21}}{2}
\][/tex]
Numerically, these solutions are approximately [tex]\( x = 0.7913 \)[/tex] and [tex]\( x = -3.7913 \)[/tex].
The inflection points [tex]\( x = D, E, F \)[/tex] are therefore [tex]\( x = -3.7913, 0, 0.7913 \)[/tex].
### Step 3: Determine Concavity
Check the intervals defined by these inflection points to determine if the function is concave up or concave down in each interval:
1. Interval [tex]\((-∞, D) = (-∞, -3.7913)\)[/tex]:
Test a point like [tex]\( x = -1000 \)[/tex].
- If [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- Here, [tex]\( f''(-1000) < 0 \)[/tex], so it is concave down.
2. Interval [tex]\((D, E) = (-3.7913, 0)\)[/tex]:
Test a point between the inflection points, such as [tex]\( x = -2 \)[/tex].
- If [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
- Here, [tex]\( f''(-2) > 0 \)[/tex], so it is concave up.
3. Interval [tex]\((E, F) = (0, 0.7913)\)[/tex]:
Choose a point like [tex]\( x = 0.5 \)[/tex].
- If [tex]\( f''(x) < 0 \)[/tex], the function is concave down.
- Here, [tex]\( f''(0.5) < 0 \)[/tex], so it is concave down.
4. Interval [tex]\((F, ∞) = (0.7913, ∞)\)[/tex]:
Test a point in this interval, such as [tex]\( x = 1000 \)[/tex].
- If [tex]\( f''(x) > 0 \)[/tex], the function is concave up.
- Here, [tex]\( f''(1000) > 0 \)[/tex], so it is concave up.
### Conclusion
Based on these results, the inflection points and concavity of the function are as follows:
- Inflection points:
[tex]\( D = -3.7913 \)[/tex], [tex]\( E = 0 \)[/tex], [tex]\( F = 0.7913 \)[/tex]
- Concavity:
- [tex]\((-∞, D)\)[/tex]: Concave down
- [tex]\((D, E)\)[/tex]: Concave up
- [tex]\((E, F)\)[/tex]: Concave down
- [tex]\((F, ∞)\)[/tex]: Concave up
