Find domain, intercepts, extrema, and concavity for

Question

f(x)=2x/(x^2-1). 

Answer 1

The Function is .

a)

The denominator x²-1 should never be equal to zero.

x = ±1

Therefore the Domain of x is {x / x ∈ R, x ≠ ±1}

b)

To find the intercepts of the function,

To find the y - intercept , we substitute x = 0.

Then y = 0. therefore y-intercept is zero.

To find the x - intercept , we substitute y = 0.

Then 2x = 0.

Therefore x-intercept is x = 0.

Therefore x - intercept point is (0,0) and y - intercept point is (0,0).

c)

The Function is

Differentiate with respect to x.

     (Using Rule of Differentiation )

To find the relative Extrema, f'(x) = 0.

1+x² = 0

x²  = -1

Therefore the value of x is imaginary values which is not considered.

There is no extrema points for f(x)

d)

Differentiate with respect to x.

Therefore to find inflection points, f''(0) = 0

if 4x=0 then x = 0

If then x = 1.7320 , -1.7320, i and -i.

Imaginary roots are not considered then

Inflection points are x = -1.732 , 0 and 1.732

Therefore the test intervals are assumed as (-∞ ,-1.732) , (-1.732 , 0), (0, 1.732)  and (1.732, ∞)

Interval        Test Value                          Sign of f''(x)                                                    Conclusion

(-∞,-1.732)  x = -2        Concave downward

(-1.732,0)    x = -1.5     Concave upward

(0,1.732)     x = 1.5           Concave downward

(1.732, ∞)      x = 2                           Concave upward.