# How do you determine if a function is concave up/ concave down if #tanx+2x #on #(-pi/2, pi/2)#?

Investigate the sign of the second derivative.

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To determine the concavity of the function ( f(x) = \tan(x) + 2x ) on the interval ( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) ), you need to find the second derivative of the function and then analyze its sign within the given interval.

First, find the first derivative ( f'(x) ) of ( f(x) ), then find the second derivative ( f''(x) ).

[ f'(x) = \sec^2(x) + 2 ]

[ f''(x) = \frac{d}{dx}(\sec^2(x) + 2) ]

[ = \frac{d}{dx}(\sec^2(x)) + \frac{d}{dx}(2) ]

[ = 2\sec^2(x)\tan(x) ]

Now, you need to determine the sign of ( f''(x) ) within the interval ( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) ). You can observe that ( \sec^2(x) ) is always positive within this interval. And ( \tan(x) ) is positive in the interval ( \left(-\frac{\pi}{2}, 0\right) ) and negative in the interval ( \left(0, \frac{\pi}{2}\right) ). Since ( \sec^2(x) ) is always positive, the sign of ( f''(x) ) depends solely on ( \tan(x) ).

In the interval ( \left(-\frac{\pi}{2}, 0\right) ), ( f''(x) ) is positive, indicating concavity up.

In the interval ( \left(0, \frac{\pi}{2}\right) ), ( f''(x) ) is negative, indicating concavity down.

Therefore, the function ( f(x) = \tan(x) + 2x ) is concave up on ( \left(-\frac{\pi}{2}, 0\right) ) and concave down on ( \left(0, \frac{\pi}{2}\right) ).

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