# Can someone please explain to me what this question is asking and how I should go about solving it: Of all the lines tangent to the curve f(x)=6/(x^2+3) find the x-coordinate of the tangent lines of minimum and maximum slope?

Here is a solution that uses calculus (not a precalculus solution).

By signing up, you agree to our Terms of Service and Privacy Policy

To find the x-coordinate of the tangent lines of minimum and maximum slope for the curve f(x) = 6/(x^2+3), we need to determine the points on the curve where the slope of the tangent line is at its minimum and maximum.

To find the slope of the tangent line, we can differentiate the function f(x) with respect to x.

Differentiating f(x) = 6/(x^2+3) using the quotient rule, we get:

f'(x) = [6*(2x)] / (x^2+3)^2

To find the x-coordinate of the tangent lines of minimum and maximum slope, we need to find the critical points of f'(x).

Setting f'(x) equal to zero and solving for x, we get:

[6*(2x)] / (x^2+3)^2 = 0

Simplifying, we have:

2x = 0

Therefore, x = 0 is the only critical point of f'(x).

To determine whether this critical point corresponds to a minimum or maximum slope, we can analyze the second derivative of f(x).

Differentiating f'(x) = [6*(2x)] / (x^2+3)^2, we get:

f''(x) = [6*(2*(x^2+3)^2) - 6*(2x)*(2*(x^2+3))*(2x)] / (x^2+3)^4

Simplifying, we have:

f''(x) = [12*(x^2+3) - 24x^2] / (x^2+3)^3

To determine the sign of f''(x) at x = 0, we substitute x = 0 into f''(x):

f''(0) = [12*(0^2+3) - 24*0^2] / (0^2+3)^3

Simplifying, we have:

f''(0) = 12/27

Since f''(0) is positive, the critical point x = 0 corresponds to a minimum slope.

Therefore, the x-coordinate of the tangent line of minimum slope is x = 0.

To find the x-coordinate of the tangent line of maximum slope, we need to consider the endpoints of the curve.

As x approaches positive or negative infinity, the value of f(x) approaches 0.

Therefore, the x-coordinate of the tangent line of maximum slope is x = positive or negative infinity.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the equation of the tangent line of #f(x)=(x-2)/(x-3)+lnx/(3x)-x# at #x=2#?
- How do you use the limit definition of the derivative to find the derivative of #f(x)=3x-7#?
- How do you find the tangent line to #y = x^2 + 3x - 4#?
- What is the average rate of change in the interval #(-4,4)#, if #f(-4)=0# and #f(4)=-5#?
- How do you find f'(x) using the definition of a derivative #f(x) =(x-6)^(2/3)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7