How do you determine whether the function #f(x)= -6 sqrt (x)# is concave up or concave down and its intervals?

Answer 1

Use calculus (the sign of the second derivative) or algebra/precalculus graphing techniques.

Calculus

In general, to investigate concavity of the graph of function #f#, we investigate the sign of the second derivative.
#f(x) = -6x^(1/2)#
Note first that the doamin of #f# is #[0,oo)#
#f'(x) = -3x^(-1/2)#
#f''(x) = 3/2 x^(-3/2) = 3/(2sqrtx^3)#
#f''(x)# is positive for all real #x#, so it is positive for all #x# in the domain of #f#.
The graph of #f# is concave up on #(0,oo)#.

(Intervals of concavity are generally given as open intervals.)

Algebra/Precalculus

The graph of the square root function looks like this:

graph{y = sqrtx [-10, 10, -5, 5]}

That graph is concave down.

Multiplying by #-6# reflects the graph across the #x# asix and stretches it vertically by a factor of #6#:

graph{y = -6sqrtx [-6.41, 25.63, -13.2, 2.82]}

The graph is concave up.

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Answer 2

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

  1. Find the first derivative ( f'(x) ) of ( f(x) ) using the power rule and chain rule. ( f'(x) = -6\frac{1}{2\sqrt{x}} = -\frac{3}{\sqrt{x}} )

  2. Find the second derivative ( f''(x) ) by differentiating ( f'(x) ) with respect to ( x ). ( f''(x) = \frac{3}{2x^{3/2}} )

Now, determine the sign of ( f''(x) ) to identify concavity:

  • If ( f''(x) > 0 ), the function is concave up.
  • If ( f''(x) < 0 ), the function is concave down.

For ( f''(x) = \frac{3}{2x^{3/2}} ), it's clear that ( f''(x) > 0 ) for all ( x > 0 ). Thus, the function ( f(x) = -6\sqrt{x} ) is concave up for all positive ( x ) values.

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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.

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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