How do you determine whether the function #f(x)= sinxcosx# is concave up or concave down and its intervals?
See the explanation section.
There are, of course other ways to write the intervals.
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To determine whether the function ( f(x) = \sin(x)  \cos(x) ) is concave up or concave down and its intervals, you need to find the second derivative of the function and then examine its sign.

Find the first derivative: [ f'(x) = \frac{d}{dx} (\sin(x)  \cos(x)) = \cos(x) + \sin(x) ]

Find the second derivative: [ f''(x) = \frac{d^2}{dx^2} (\cos(x) + \sin(x)) = \sin(x) + \cos(x) ]

Determine the intervals where ( f''(x) > 0 ) for concave up and where ( f''(x) < 0 ) for concave down.

Solve ( \sin(x) + \cos(x) > 0 ) for concave up: [ \sin(x) + \cos(x) > 0 ] [ \cos(x) > \sin(x) ]

Analyze the sign of ( \cos(x)  \sin(x) ) within the period ( 0 \leq x < 2\pi ) or ( 0^\circ \leq x < 360^\circ ) to determine the intervals where the function is concave up.

Repeat the process to determine the intervals where the function is concave down by analyzing where ( f''(x) < 0 ).
By following these steps, you can identify the intervals where the function ( f(x) = \sin(x)  \cos(x) ) is concave up or concave down.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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