# How do you find the inflection points for the function #f(x)=xsqrt(5-x)#?

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To find the inflection points for the function ( f(x) = x\sqrt{5 - x} ), follow these steps:

- Find the second derivative of the function ( f(x) ).
- Set the second derivative equal to zero and solve for ( x ).
- Determine the corresponding ( y ) values for the ( x ) values found in step 2.

Let's go through these steps:

- First, find the first derivative of ( f(x) ) using the product rule:

[ f'(x) = \frac{d}{dx}[x] \cdot \sqrt{5 - x} + x \cdot \frac{d}{dx}[\sqrt{5 - x}] ]

[ f'(x) = 1 \cdot \sqrt{5 - x} + x \cdot \frac{d}{dx}[\sqrt{5 - x}] ]

[ f'(x) = \sqrt{5 - x} - \frac{x}{2\sqrt{5 - x}} ]

- Next, find the second derivative of ( f(x) ):

[ f''(x) = \frac{d}{dx}\left[\sqrt{5 - x} - \frac{x}{2\sqrt{5 - x}}\right] ]

[ f''(x) = -\frac{1}{2}\cdot\frac{d}{dx}\left[(5 - x)^{\frac{1}{2}}\right] - \frac{1}{2\sqrt{5 - x}} + \frac{x}{4}\cdot\frac{d}{dx}\left[(5 - x)^{-\frac{1}{2}}\right] ]

[ f''(x) = -\frac{1}{2}\cdot\left[-\frac{1}{2}(5 - x)^{-\frac{3}{2}}\right] - \frac{1}{2\sqrt{5 - x}} - \frac{x}{4}\cdot\left[-\frac{1}{2}(5 - x)^{-\frac{3}{2}}\right] ]

[ f''(x) = \frac{1}{4}(5 - x)^{-\frac{3}{2}} - \frac{1}{2\sqrt{5 - x}} + \frac{x}{8}(5 - x)^{-\frac{3}{2}} ]

[ f''(x) = \frac{1}{4}(5 - x)^{-\frac{3}{2}} + \frac{x}{8}(5 - x)^{-\frac{3}{2}} - \frac{1}{2\sqrt{5 - x}} ]

[ f''(x) = \frac{1 + 2x - 5x}{8\sqrt{5 - x}(5 - x)} ]

[ f''(x) = \frac{1 - 3x}{8\sqrt{5 - x}(5 - x)} ]

- Set ( f''(x) ) equal to zero and solve for ( x ):

[ \frac{1 - 3x}{8\sqrt{5 - x}(5 - x)} = 0 ]

[ 1 - 3x = 0 ]

[ x = \frac{1}{3} ]

- Determine the corresponding ( y ) value:

[ f\left(\frac{1}{3}\right) = \frac{1}{3}\sqrt{5 - \frac{1}{3}} = \frac{1}{3}\sqrt{\frac{14}{3}} = \frac{\sqrt{14}}{3} ]

Therefore, the inflection point is ( \left(\frac{1}{3}, \frac{\sqrt{14}}{3}\right) ).

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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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