# Is #f(x)=1-x-e^(-3x)/x# concave or convex at #x=4#?

Let's take some derivatives!

Now let x = 4.

Observe that the exponential is always positive. The numerator of the fraction is negative for all positive values of x. The denominator is positive for positive values of x.

Draw your conclusion about concavity.

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To determine whether ( f(x) = 1 - x - \frac{e^{-3x}}{x} ) is concave or convex at ( x = 4 ), we need to examine the second derivative of the function at that point. If the second derivative is positive, the function is convex; if it's negative, the function is concave.

First, find the first derivative of ( f(x) ): [ f'(x) = -1 + \frac{3e^{-3x}}{x^2} + \frac{e^{-3x}(3x+1)}{x^2} ]

Then, find the second derivative: [ f''(x) = \frac{9e^{-3x}}{x^3} - \frac{6e^{-3x}}{x^2} - \frac{6e^{-3x}(3x+1)}{x^3} + \frac{2e^{-3x}(3x+1)}{x^2} ]

Now, evaluate ( f''(4) ) to determine the concavity or convexity at ( x = 4 ).

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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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