Calculus problem

Question

Determine the intervals where f is increasing and where f is decreasing (written in interval notation), and identify all relative extrema (write these as ordered pairs).

f(x)=2x+8/x-5

Answer 1

You just have to find out where f'(x)>0, f'(x)<0 and f'(x) = 0 or where f'(x) is undefined.

f(x)=2x+8/x-5

So f'(x) = 2-8/x^2

2-8/x^2 >0 then 2>8/x^2 then x^2 >4, x>2 or x<-2... This is where f is increasing.

Now solve the other inequalities and equations.

Answer 2

f(x)=2x+8/x-5

I'm going to assume this means f(x)=2x+(8/x)-5. There is otherwise a small chance it could be misinterpreted f(x)=2x+8/(x-5).

We know that when a function f(x) is increasing it has a positive slope. So whenever f'(x)>0 the function is increasing. Likewise, when f(x) is decreasing it has a negative slope. So whenever f'(x)<0 the function is decreasing. When f'(x)=0, we have a local extreme point.

f(x)=2x+8x^-1-5 so

f'(x)=2-8x^-2

when f'(x)=0, 8x^-2=2, x^-2=1/2, x²=2, x=±√2

Now the second derivative provides the nature of the extreme points. If f''(x)>0, the curve is concave up (a minimum). If f''(x)<0, it's concave down (a maximum).

f''(x)=16x^-3=16/x³

f''(-√2)=16/(-√2)³<0 so there's a local maximum at x=-√2

f(-√2)=2(-√2)+8/(-√2)-5=-2√2-8/√2-5=-13.48

So a local maximum is at coordinate (-√2,-13.48)

f''(√2)=16/(√2)³>0 so there's a local minimum at x=√2

f(√2)=2(√2)+8/(√2)-5=2√2+8/√2-5=3.48

So a local minimum is at coordinate (√2,3.48)

f(x) has a vertical asymptote at x=0 due to the 8/x term.

So the intervals we are concerned with are (-∞,-√2], (-√2,0), (0,√2], (√2,∞)

Since there's a local maximum at x=-√2 the curve is increasing over interval (-∞,-√2) and decreasing over interval (-√2,0).

Since there's a local minimum at x=√2 the curve is decreasing over interval (0,√2) and increasing over interval (√2,∞)

If anyone needs help graphing functions with or without calculus, I can help.

Answer 3

Do you mean:

f(x) = 2x + 8/x - 5 ?

OR:

f(x) = (2x + 8) / (x - 5) ?

In either case, first find the derivative, f'(x)

"Determine the intervals where f is increasing and where f is decreasing (written in interval notation)"

If f(x) is increasing, then f'(x) will be positive, and--if decreasing--f'(x) will be negative

At the "relative extrema", f'(x) will be zero

Use a graphing calculator or online tool the help visualize. eg DESMOS:

https://www.desmos.com/calculator