# For what values of x is #f(x)= e^x/(5x^2 +1# concave or convex?

The function is convex in the intervals

The function is

Calculate the first and second derivatives

The first derivative is the derivative of a quotient

The second derivative is the derivative of a quotient

Therefore,

The points of inflections are when

graph{25x^4-100x^3+160x^2-20x-9 [-3.465, 3.464, -1.73, 1.734]}

Let's build a variation chart to determine the concavities

graph{e^x/(5x^2+1) [-3.465, 3.464, -1.73, 1.734]}

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The function f(x) = e^x / (5x^2 + 1) is convex for all values of x.

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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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- What are the points of inflection, if any, of #f(x) = 5x^3 + 30x^2 - 432x #?
- How do you sketch the graph #y=x^2/(1+x^2)# using the first and second derivatives?

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