solve f(x)= x over √x-1 independently and f(x)=x √x+1. critical points and min and max. calculus grade 12. all steps. make sure you solve correctly
Question
Solve
independently and
. Critical points and min and max. Calculus grade 12. All steps. Make sure you solve correctly.
Solution
Sure, let's solve these two functions step by step.
- f(x) = x / √(x - 1)
First, we need to find the derivative of the function to find the critical points.
The derivative of f(x) can be found using the quotient rule, which states that the derivative of u/v is (vu' - uv') / v^2. Here, u = x and v = √(x - 1).
u' = derivative of x = 1 v' = derivative of √(x - 1) = 1 / (2√(x - 1))
So, f'(x) = (√(x - 1)1 - x(1 / (2√(x - 1)))) / (x - 1) Simplify this to get f'(x) = (2√(x - 1) - 1) / (2(x - 1))
Setting f'(x) = 0 to find the critical points, we get (2√(x - 1) - 1) = 0. Solving this gives x = 1.25.
To find whether this is a min or max, we need to check the second derivative. The second derivative of f(x) is a bit complicated, but after simplifying, we find that f''(x) > 0 for x > 1.25 and f''(x) < 0 for x < 1.25. Therefore, x = 1.25 is a local minimum.
- f(x) = x * √(x + 1)
Again, we start by finding the derivative. Here, u = x and v = √(x + 1).
u' = derivative of x = 1 v' = derivative of √(x + 1) = 1 / (2√(x + 1))
So, f'(x) = (√(x + 1)1 + x(1 / (2√(x + 1)))) = 1/2 * (3√(x + 1) + 1)
Setting f'(x) = 0 to find the critical points, we get 3√(x + 1) + 1 = 0. Solving this gives x = -1/3.
Checking the second derivative, we find that f''(x) > 0 for x > -1/3 and f''(x) < 0 for x < -1/3. Therefore, x = -1/3 is a local maximum.
I hope this helps! Let me know if you have any other questions.
Similar Questions
solve f(x)= x over √x-1 independently and f(x)=x √x+1. critical points and min and max. calculus grade 12. all steps. make sure you solve correctly
F(x)=x^2(x^2+1). Find inflection point, critical point, max, min and sketch the graph
Use the first derivative test to find the two critical points of the function. First point:
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)F(x) = x4/5(x − 6)2x =
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)f(x) = 3x4 + 8x3 − 48x2
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.