Find the inflection points for f(x) = 12x^5 + 45x^4 - 360x^3 + 7
f'(x) = 60x^4 + 180x^3 - 1080x^2
f''(x) = 240x^3 + 540x^2 - 2160x
= 0 for points of inflection
240x^3 + 540x^2 - 2160x = 0
divide by 60
4x^3 + 9x^2 - 36x = 0
x(4x^2 + 9x - 36) = 0
Can you take it from there ?
There are 3 solutions for x
I got 0 for one of my inflections since a 0 in the place of the "x"(4x^2 + 9x - 36) would make it zero. But I'm not sure what would need to go into the other x's to make it 0. I tried 4/9 9/36 and 36/9 but I couldn't get it.
Oh, okay. I found out I had to use the quadratic formula. Thanks for the help!
To find the inflection points of a function, we need to follow these steps:
1. Find the second derivative of the function.
2. Determine the values of x for which the second derivative equals zero or is undefined.
3. Substitute the values obtained in step 2 into the original function to find the corresponding y-values.
Let's begin by finding the second derivative of the function f(x) = 12x^5 + 45x^4 - 360x^3 + 7.
Step 1: Compute the first derivative of f(x)
f'(x) = 60x^4 + 180x^3 - 1080x^2
Step 2: Compute the second derivative of f(x)
f''(x) = 240x^3 + 540x^2 - 2160x
Now, we need to solve the equation f''(x) = 0 to find the values of x where the second derivative equals zero.
240x^3 + 540x^2 - 2160x = 0
We can factor out common terms:
240x(x^2 + 9/2x - 9) = 0
Now we have two factors:
Factor 1: x = 0 (from 240x = 0)
Factor 2: x^2 + 9/2x - 9 = 0
We can solve the quadratic equation x^2 + 9/2x - 9 = 0 using factoring, completing the square, or quadratic formula. Solving this equation, we get two additional values for x, which are approximately x = -9.461 and x = 0.961.
Hence, the values of x for which the second derivative equals zero or is undefined are x = 0, x = -9.461, and x = 0.961.
Step 3: Substitute these x-values into the original function f(x) to find the corresponding y-values.
By substituting x = 0 into f(x), we get f(0) = 7.
By substituting x = -9.461 into f(x), we get f(-9.461) = -612,955.428.
By substituting x = 0.961 into f(x), we get f(0.961) = -1,715.366.
Therefore, the inflection points of the function f(x) = 12x^5 + 45x^4 - 360x^3 + 7 are (0, 7), (-9.461, -612,955.428), and (0.961, -1,715.366).