# How do you sketch the curve #y=(x+1)/sqrt(5x^2+35)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

see below

graph{(x+1)/(sqrt(5x^2+35)) [-10, 10, -5, 5]}

function doesn't have minimum

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To sketch the curve (y = \frac{x+1}{\sqrt{5x^2 + 35}}), we first find the critical points by taking the derivative, set it to zero to find the local extrema, and then find the inflection points and asymptotes. Finally, we plot the curve by incorporating these points.

**1. Find the Derivative:**

[y' = \frac{d}{dx} \left( \frac{x+1}{\sqrt{5x^2 + 35}} \right)]

[y' = \frac{\sqrt{5x^2 + 35} - (x+1)(\frac{1}{2})(5x^2 + 35)^{-\frac{1}{2}}(10x)}{5x^2 + 35}]

[y' = \frac{(5x^2 + 35) - (x+1)(10x)}{2(5x^2 + 35)^\frac{3}{2}}]

[y' = \frac{5x^2 + 35 - 10x^2 - 10x}{2(5x^2 + 35)^\frac{3}{2}}]

[y' = \frac{-5x^2 - 10x + 35}{2(5x^2 + 35)^\frac{3}{2}}]

**2. Find the Critical Points:**

Set (y' = 0) to find critical points.

[-5x^2 - 10x + 35 = 0]

[5x^2 + 10x - 35 = 0]

[x^2 + 2x - 7 = 0]

[(x + 3)(x - 1) = 0]

[x = -3, 1]

**3. Find the Local Extrema:**

Use the second derivative test to find local extrema.

[y'' = \frac{d^2}{dx^2} \left( \frac{x+1}{\sqrt{5x^2 + 35}} \right)]

[y'' = \frac{d}{dx} \left( \frac{-5x^2 - 10x + 35}{2(5x^2 + 35)^\frac{3}{2}} \right)]

[y'' = \frac{-10x - 10}{2(5x^2 + 35)^\frac{3}{2}} - \frac{-5x^2 - 10x + 35}{2(5x^2 + 35)^\frac{5}{2}}(10x)]

[y'' = \frac{-10x - 10}{2(5x^2 + 35)^\frac{3}{2}} + \frac{(5x^2 + 10x - 35)(10x)}{2(5x^2 + 35)^\frac{5}{2}}]

**4. Find the Inflection Points:**

Set (y'' = 0) to find inflection points.

**5. Find Asymptotes and Intercepts:**

To find asymptotes and intercepts, analyze the behavior of the function as (x) approaches infinity and negative infinity. Also, find the (y)-intercept by setting (x = 0).

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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.

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