Find the positive x-value where the function f(x)= 1/(-4x^2-3) has an inflection point.?

Question

One last one, help me with this I promise to quit asking

Answer 1

Find the derivative of your equation and set it to zero... that will be a turning point on your original equation.

This is because the derivative describes the slope of your equation and at an inflection point it is changing from positive to negative or the other way... the slope (or value of the derivative) is zero.

Aloha

Answer 2

first, rewrite as f(x)=-1/(4x^2+3).

Now proceed as in the last problem.

f'(x) = 8x / (4x^2+3)^2, by the power rule and chain rule.

By the quotient rule (and chain rule again),

f''(x) = [(4x^2+3)^2(8) - (8x)(2)(4x^2+3)(8x)] / (4x^2+3)^4.

You want to see when the numerator is 0, because then f''(x) will be 0. The numerator is

(4x^2+3)^2(8) - (8x)(2)(4x^2+3)(8x)

Factor out things that are always positive: 8(4x^2+3). You're left with solving

4x^2+3 - 16x^2 = 0.

This simplifies to

3 - 12x^2 = 0, or

1 - 4x^2 = 0.

The solutions are x=1/2 or x=-1/2, so the positive x-value is 1/2.

NOTE: the answer below refers to a local maximum or minimum, not an inflection point, which is a change in the concavity of the function's graph, not a change in the sign of its slope.

Answer 3

first, rewrite as f(x)=-1/(4x^2+3).

Now proceed as in the last problem.

f'(x) = 8x / (4x^2+3)^2, by the power rule and chain rule.

By the quotient rule (and chain rule again),

f''(x) = [(4x^2+3)^2(8) - (8x)(2)(4x^2+3)(8x)] / (4x^2+3)^4.

You want to see when the numerator is 0, because then f''(x) will be 0. The numerator is

(4x^2+3)^2(8) - (8x)(2)(4x^2+3)(8x)

Factor out things that are always positive: 8(4x^2+3). You're left with solving

4x^2+3 - 16x^2 = 0.

This simplifies to

3 - 12x^2 = 0, or

1 - 4x^2 = 0.

The solutions are x=1/2 or x=-1/2, so the positive x-value is 1/2.

NOTE: the answer below refers to a local maximum or minimum, not an inflection point, which is a change in the concavity of the function's graph, not a change in the sign of its slope.

Answer 4

Definition of Inflection Point

A point at which the graph of a function changes concavity is called an inflection point.
This may be a point where the second derivative:
does not exist, or
equals zero.


f(x) =-1/(4x^2+3)
f"(x) = 8x/(4x^2+3)^2
f''(x) = (8(4x^2+3)^2 - 8x*2*8x(4x^2+3))/(4x^2+3)^4 = 0

The 2nd derivative has one positive and one negative root

Answer 5

The solutions of James L and Jim M are basically marvelous, yet as quickly as you resolve f"(x) = 0, you could desire to verify if the 2d by-product differences sign simply by fact the curve passes by each and every of the factors you discovered.

Answer 6

do the first derivative, then the second and do a number line thing