Question

let f(x)=x^4 - 6x^3+12x^2. find (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the coordinates of all inflection points. (a) f is increasing on the interval(s) (b) f is decreasing on the interval(s) (c) f is concave up on the open interval(s) (d) f is concave down on the open interval(s) (e) the x - coordinate(s) of the points of inflection are notes: in the first four boxes, your answer should either be a single interval, such as 0,1), a comma - separated list of intervals, such as (-inf,2),(3,4, or the word
one\. in the last box, your answer should be a comma - separated list of x values or the word
one\. submit answer next item

Explanation:

1. First, find the first - derivative of \(y = f(x)=x^{4}-6x^{3}+12x^{2}\):

• Using the power rule \((x^n)^\prime=nx^{n - 1}\), we have \(y^\prime=f^\prime(x)=4x^{3}-18x^{2}+24x = 2x(2x^{2}-9x + 12)\).
• To find the critical points, set \(y^\prime = 0\). Since the discriminant of the quadratic \(2x^{2}-9x + 12\) is \(\Delta=b^{2}-4ac=(-9)^{2}-4\times2\times12=81 - 96=-15\lt0\), the only real - root of \(y^\prime = 0\) is \(x = 0\).
• Now, use the test - point method. Choose test points in the intervals \((-\infty,0)\) and \((0,\infty)\). Let's take \(x=-1\) for the interval \((-\infty,0)\): \(y^\prime(-1)=2\times(-1)\times(2 + 9+12)=-46\lt0\). Let's take \(x = 1\) for the interval \((0,\infty)\): \(y^\prime(1)=2\times1\times(2 - 9 + 12)=10\gt0\).
• (a) Intervals of increase and decrease:
• A function \(y = f(x)\) is increasing when \(y^\prime\gt0\) and decreasing when \(y^\prime\lt0\).
• The function \(f(x)\) is decreasing on the interval \((-\infty,0)\) and increasing on the interval \((0,\infty)\).

1. Then, find the second - derivative of \(y = f(x)\):

• Differentiate \(y^\prime=4x^{3}-18x^{2}+24x\) with respect to \(x\). Using the power rule, \(y^{\prime\prime}=f^{\prime\prime}(x)=12x^{2}-36x + 24=12(x^{2}-3x + 2)=12(x - 1)(x - 2)\).
• (b) Intervals of concavity:
• Set \(y^{\prime\prime}=0\), then \(12(x - 1)(x - 2)=0\), so \(x = 1\) and \(x = 2\).
• Choose test points in the intervals \((-\infty,1)\), \((1,2)\), and \((2,\infty)\).
• For the interval \((-\infty,1)\), let \(x = 0\), then \(y^{\prime\prime}(0)=12\times(0 - 1)\times(0 - 2)=24\gt0\).
• For the interval \((1,2)\), let \(x=\frac{3}{2}\), then \(y^{\prime\prime}(\frac{3}{2})=12\times(\frac{3}{2}-1)\times(\frac{3}{2}-2)=12\times\frac{1}{2}\times(-\frac{1}{2})=-3\lt0\).
• For the interval \((2,\infty)\), let \(x = 3\), then \(y^{\prime\prime}(3)=12\times(3 - 1)\times(3 - 2)=24\gt0\).
• The function \(f(x)\) is concave up on the intervals \((-\infty,1)\cup(2,\infty)\) and concave down on the interval \((1,2)\).
• (c) Inflection points:
• Inflection points occur where \(y^{\prime\prime}\) changes sign. Since \(y^{\prime\prime}=12(x - 1)(x - 2)\) changes sign at \(x = 1\) and \(x = 2\), and \(f(1)=1^{4}-6\times1^{3}+12\times1^{2}=1 - 6 + 12 = 7\), \(f(2)=2^{4}-6\times2^{3}+12\times2^{2}=16-48 + 48 = 16\). The inflection points are \((1,7)\) and \((2,16)\), and the \(x\) - coordinates of the inflection points are \(x = 1,2\).

Step1: Find the first - derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), \(f^\prime(x)=4x^{3}-18x^{2}+24x = 2x(2x^{2}-9x + 12)\).

Step2: Determine critical points

Set \(f^\prime(x)=0\). The quadratic \(2x^{2}-9x + 12\) has no real roots (\(\Delta\lt0\)), and \(x = 0\) is the critical point. Use test - points to find intervals of increase and decrease.

Step3: Find the second - derivative

Differentiate \(f^\prime(x)\) to get \(f^{\prime\prime}(x)=12x^{2}-36x + 24=12(x - 1)(x - 2)\).

Step4: Determine inflection points and intervals of concavity

Set \(f^{\prime\prime}(x)=0\) to find \(x = 1\) and \(x = 2\). Use test - points to find intervals of concavity.

Answer:

(a) Increasing interval: \((0,\infty)\); Decreasing interval: \((-\infty,0)\)
(b) Concave up intervals: \((-\infty,1),(2,\infty)\); Concave down interval: \((1,2)\)
(c) \(x\) - coordinates of inflection points: \(1,2\)