The inflection point of the curve y = x4 − 8x3 + 24x2 + 7x − 3 is:
I've been getting confused on how to do this but I keep getting (0,3)
The response above is correct, but when you check your answer for any sign change, you'll see that at x=2, it's actually not an inflection point. The concavity does not change.
The answer is: There is no point of inflection
(at least that's the answer on apex)
y'=4x^3-24x^2+48x+7
y"=12x^2-48x+48
set y" to zero, and solve for x
x^2-4x+4=0
(x-2)^2=0
x=2
y= use original equation to find y.
Well, it seems like you're trying to find the inflection point of a curve. I hate to burst your bubble, but (0,3) is not the correct answer. That's just some point on the curve, not the inflection point.
But hey, don't worry! Finding the inflection point can be a tricky business. It's like trying to find a hidden banana in a room full of oranges.
To find the inflection point, you'll need to take the second derivative of the function y = x^4 - 8x^3 + 24x^2 + 7x - 3. Then, set the second derivative equal to zero and solve for x.
After solving that mathematical puzzle, you should find that the inflection point is somewhere else. Keep searching, my friend! You'll find it eventually. And when you do, throw a party to celebrate! Trust me, inflection points are just the life of the party.
To find the inflection point of a curve, we need to find the x-coordinate where the concavity of the curve changes. The concavity changes when the second derivative of the equation changes sign. Let's go step-by-step to find the inflection point of the given curve y = x^4 - 8x^3 + 24x^2 + 7x - 3.
Step 1: Find the first derivative.
Differentiating y = x^4 - 8x^3 + 24x^2 + 7x - 3 with respect to x, we get:
y' = 4x^3 - 24x^2 + 48x + 7.
Step 2: Find the second derivative.
Differentiating y', we get:
y'' = 12x^2 - 48x + 48.
Step 3: Set y'' equal to zero and solve for x.
12x^2 - 48x + 48 = 0.
Dividing by 12, we get:
x^2 - 4x + 4 = 0.
Factoring, we have:
(x - 2)^2 = 0.
Taking the square root, we get:
x - 2 = 0.
Solving for x, we find:
x = 2.
Step 4: Plug the x-coordinate into the original equation to find the y-coordinate of the inflection point.
Plugging x = 2 into y = x^4 - 8x^3 + 24x^2 + 7x - 3, we get:
y = (2)^4 - 8(2)^3 + 24(2)^2 + 7(2) - 3
= 16 - 64 + 96 + 14 - 3
= 59.
Therefore, the inflection point of the given curve is (2, 59).
To find the inflection point of a curve, we need to determine the x-coordinate where the concavity changes. In other words, the inflection point occurs where the curve changes from being concave upwards to concave downwards, or vice versa.
To find the inflection point of the curve y = x^4 − 8x^3 + 24x^2 + 7x − 3, we will follow these steps:
1. Start by finding the second derivative of the curve. Since the given equation is y = x^4 − 8x^3 + 24x^2 + 7x − 3, we can differentiate it twice using the power rule:
First derivative:
dy/dx = 4x^3 - 24x^2 + 48x + 7
Second derivative:
d^2y/dx^2 = 12x^2 - 48x + 48
2. Now, we need to solve the equation d^2y/dx^2 = 0 to find the x-coordinate of the inflection point. In this case:
12x^2 - 48x + 48 = 0
3. To simplify the equation, we can divide all terms by 12:
x^2 - 4x + 4 = 0
4. Solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, the equation can be easily factored as:
(x - 2)^2 = 0
5. By solving for x, we find:
x - 2 = 0
x = 2
6. So, the x-coordinate of the inflection point is x = 2.
To find the y-coordinate of the inflection point, substitute the obtained x-coordinate into the original equation:
y = (2)^4 − 8(2)^3 + 24(2)^2 + 7(2) − 3
y = 16 - 64 + 96 + 14 - 3
y = 59
Therefore, the inflection point of the curve y = x^4 − 8x^3 + 24x^2 + 7x − 3 is (2, 59).