How do you find the concavity for #f(x)= -6 sqrt (x)#?

Answer 1

Evelyn Agard

Concave for all #x# with #x>0#

Writing

#f(x)=-6x^(1/2)#

#f'(x)=-3x^(-1/2)#

#f''(x)=3/2*x^(-3/2)>0#

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Answer 2

Charles Arocha

To find the concavity of ( f(x) = -6\sqrt{x} ), you need to find the second derivative of the function and then determine its sign.

First, find the first derivative of ( f(x) ) using the power rule for differentiation:

( f'(x) = -6 \cdot \frac{1}{2\sqrt{x}} = -\frac{6}{2\sqrt{x}} = -\frac{3}{\sqrt{x}} )

Next, find the second derivative by differentiating ( f'(x) ):

( f''(x) = \frac{d}{dx}(-\frac{3}{\sqrt{x}}) = \frac{3}{2x^{3/2}} )

Now, to determine the concavity, you need to examine the sign of ( f''(x) ). Since ( x ) is positive (as ( \sqrt{x} ) requires non-negative inputs), ( f''(x) ) is always positive. Thus, the concavity of ( f(x) = -6\sqrt{x} ) is upward or concave upwards throughout its domain.

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Answer from HIX Tutor

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.