Consider the function f(x)=12x^5+30x^4–160x^3+5 .
f(x) has inflection points at (reading from left to right) x=D , E , and F
D=
E=
F=
To find the inflection points of a function, we need to determine the values of x where the concavity of the graph changes. In other words, we're looking for the x-values where the second derivative of the function equals zero or does not exist.
Step 1: Find the first derivative of f(x):
f'(x) = 60x^4 + 120x^3 - 480x^2
Step 2: Find the second derivative of f(x):
f''(x) = 240x^3 + 360x^2 - 960x
Step 3: Set f''(x) equal to zero and solve for x:
240x^3 + 360x^2 - 960x = 0
Step 4: Factor out common terms:
x(240x^2 + 360x - 960) = 0
Step 5: Set each factor equal to zero and solve for x:
x = 0 (inflection point D)
240x^2 + 360x - 960 = 0
Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula:
In this case, the quadratic equation does not factor easily. So, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 240, b = 360, and c = -960
x = (-360 ± √(360^2 - 4 * 240 * -960)) / (2 * 240)
x = (-360 ± √(129600 + 921600)) / 480
x = (-360 ± √1051200) / 480
x = (-360 ± √(2^6 * 3^2 * 5^2 * 181)) / 480
x = (-360 ± 2 * 3 * 5 * √(181)) / 480
x = (-6 ± √(181)) / 4
So, we have two possible values for x:
x = (-6 + √(181)) / 4 (inflection point E)
x = (-6 - √(181)) / 4 (inflection point F)
Therefore, the inflection points of the function f(x) = 12x^5 + 30x^4 - 160x^3 + 5 are:
D = 0
E = (-6 + √(181)) / 4
F = (-6 - √(181)) / 4
To find the inflection points of the function f(x) = 12x^5 + 30x^4 - 160x^3 + 5, we need to find the values of x where the concavity changes.
Let's start by finding the second derivative of f(x).
f'(x) = 60x^4 + 120x^3 - 480x^2
Now let's find the second derivative:
f''(x) = 240x^3 + 360x^2 - 960x
To find the inflection points, we need to find the values of x where f''(x) = 0.
240x^3 + 360x^2 - 960x = 0
Now we can factor out x:
x(240x^2 + 360x - 960) = 0
Setting each factor equal to zero gives us:
x = 0
240x^2 + 360x - 960 = 0
Applying the quadratic formula to solve for x:
x = (-360 ± sqrt(360^2 - 4 * 240 * -960))/(2 * 240)
Simplifying:
x = (-360 ± sqrt(129600 + 921600))/480
x = (-360 ± sqrt(1051200))/480
x = (-360 ± 1024.69)/480
x ≈ -1.95 or x ≈ 2.12
Therefore, the inflection points of f(x) are approximately:
D = -1.95
E = 2.12
F = There is no inflection point at F, only D and E.