I have 2 questions for calculus homework, need the answers please, I am on my last attempt for submitting

Question

Question 1: Find the high-order derivative.

f"(x) = 4 √ x-6,

Radical is over x-6^^^

f^(5)(x) = ___________

Question 2:

f(x) = (x + 2)(x − 2)(x + 7)(x − 7)

f"(x) = 12x^2-106

Solve the equation f''(x) = 0.

 (Enter your answers as a comma-separated list.)

x = ___________

 

 

Answer 1

f"(x) = 4(x - 6)^1/2

f'''(x) = 2(x - 6)^-1/2

f''''(x) = -(x - 6)^-3/2

f'''''(x) = (3/2)(x - 6)^-5/2

----------------------------

f"(x) = 12x² - 106 = 0

12x² = 106

x² = 106/12 = 53/6

x = ±√(53/6) = ±√53 / √6 = ±√318 / 6 ≈ ± 2.97

Answer 2

Hey Antonio,

Hope you are doing well. Instead of directly giving you the answer, I would offer a few pointers on how you can approach the problem, but you will have to exert the effort on your end to actually do the computation.

For the first one, you know that f''(x) = 4 * sqrt(x-6). Finding f⁽⁵⁾(x) means find the 5th derivative. You already are at the 2nd derivative. I'd recommend differentiating 4 * sqrt(x-6) a few more times and see if you come across a pattern in the numbers (or you could differentiate 3 more times and get the answer that way too

For the 2nd one, you already know that f''(x) = 12x^2 - 106. All we need to do is solve 12x^2 - 106 = 0. Isolate the equation for x^2 and then solve for x. Remember what happens when you take the square root of a positive number (consider both cases)