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…

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 \none\. in the last box, your answer should be a comma - separated list of x values or the word \none\. submit answer next item

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 \none\. in the last box, your answer should be a comma - separated list of x values or the word \none\. submit answer next item

Answer

  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)).
  2. 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).

Explanation:

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)