(1 point) Find the inflection points of f(x)=4x4+22x3−18x2+10. (Give your answers as a comma separated list, e.g., 3,-2.)
(1 point) Is the graph of y=sin(x4) increasing or decreasing when x=14?
(enter increasing, decreasing, or neither).
Is it concave up or concave down?
(enter up, down, or neither).
ok, I'll help on this one too. But I'm sure your text explains this all in detail.
an inflection point occurs (usually) when f"(x) = 0 and f'(x) ≠ 0
the graph is concave up if f"(x) > 0
concave down if f"(x) < 0
for y = 4x^4+22x^3−18x^2+10
y' = 16x^3+66x^2-36x = 2(8x^3+33x^2-18x)
y" = 2(24x^2+66x-18) = 12(4x^2+11x-3) = 12(x+3)(4x-1)
so, there are inflection points at -3, 1/4 since f' ≠ 0 there
for y = sin(x^4)
y' = 4x^3 cos(x^4)
y" = 4x^2(3cos(x^4) - 4x^4 sin(x^4))
y'(14) = 4*14^3 cos(14^4) > 0 so f is increasing
y"(14) = 4*14^2(3cos(14^4) - 4*14^4 sin(14^4)) < 0 so the graph is concave down
To find the inflection points of the function f(x), we need to find the values of x where the concavity of the function changes. This occurs when the second derivative of the function equals zero or does not exist.
Step 1: Find the first derivative of f(x):
f'(x) = 16x^3 + 66x^2 - 36x
Step 2: Find the second derivative of f(x):
f''(x) = 48x^2 + 132x - 36
Step 3: Set f''(x) equal to zero and solve for x:
48x^2 + 132x - 36 = 0
Step 4: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation, a = 48, b = 132, and c = -36.
x = (-132 ± √(132^2 - 4(48)(-36))) / (2(48))
Simplifying further:
x = (-132 ± √(17424 + 6912)) / (96)
x = (-132 ± √24336) / 96
x = (-132 ± 156) / 96
This gives two possible solutions:
x1 = (-132 + 156) / 96 = 24 / 96 = 1/4 = 0.25
x2 = (-132 - 156) / 96 = -288 / 96 = -3
So, the inflection points of f(x) are x = 0.25 and x = -3.
To find the inflection points of a function, we need to find the points where the concavity changes. In other words, we're looking for the points where the second derivative of the function changes sign.
Let's start by finding the second derivative of the function f(x).
Given f(x) = 4x^4 + 22x^3 − 18x^2 + 10, let's find the first derivative:
f'(x) = 16x^3 + 66x^2 - 36x
Now, let's find the second derivative by taking the derivative of f'(x):
f''(x) = 48x^2 + 132x - 36
To find the inflection points, we need to solve the equation f''(x) = 0. Let's set f''(x) equal to zero and solve for x:
48x^2 + 132x - 36 = 0
Now we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 48, b = 132, and c = -36. Plugging these values into the formula, we get:
x = (-132 ± √(132^2 - 4*48*(-36))) / (2*48)
Simplifying further, we get:
x = (-132 ± √(17424 + 6912)) / 96
x = (-132 ± √(24336)) / 96
x = (-132 ± 156) / 96
This gives us two possible solutions for x:
x1 = (-132 + 156) / 96 = 24 / 96 = 1/4
x2 = (-132 - 156) / 96 = -288 / 96 = -3
Therefore, the inflection points of the function f(x) = 4x^4 + 22x^3 − 18x^2 + 10 are x = 1/4 and x = -3.