How do you determine whether the function #f(x)=(2x-3) / (x^2)# is concave up or concave down and its intervals?

Answer 1

#f(x)=(2x-3)/(x^2)# is concave up when #x > 9/2# and concave down when #x < 9/2# (and #x!=0#).

By the Quotient Rule, for #x!=0#, the first derivative is #f'(x)=(x^2*2-(2x-3)*2x)/(x^4)=(-2x^2+6x)/(x^4)=(-2x+6)/(x^3)#
and the second derivative is #f''(x)=(x^3*(-2)-(-2x+6)*3x^2)/(x^6)=(4x^3-18x^2)/(x^6)=(4x-18)/(x^4)#
Since #x^4\geq 0# for all #x#, the sign of #f''(x)# is the same as the sign of its numerator #4x-18#. This expression is positive when #4x>18\Leftrightarrow x > 9/2# and negative when #4x < 18 \Leftrightarrow x < 9/2#.
The value #x=9/2# is the first coordinate of the unique inflection point of the graph of #f#. The second coordinate is #f(9/2) = (9-3)/((9/2)^2)=6/(81/4)=24/81=8/27#
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Answer 2
To determine the concavity of the function \( f(x) = \frac{2x - 3}{x^2} \) and its intervals, we need to find its second derivative and then analyze its sign. The function will be concave up where the second derivative is positive and concave down where it is negative. First, find the first derivative: \[ f'(x) = \frac{d}{dx} \left( \frac{2x - 3}{x^2} \right) \] \[ f'(x) = \frac{(2)(x^2) - (2x - 3)(2x)}{(x^2)^2} \] \[ f'(x) = \frac{2x^2 - 4x^2 + 6x}{x^4} \] \[ f'(x) = \frac{-2x^2 + 6x}{x^4} \] \[ f'(x) = \frac{-2x(x - 3)}{x^4} \] \[ f'(x) = \frac{-2(x - 3)}{x^3} \] Now, find the second derivative: \[ f''(x) = \frac{d}{dx} \left( \frac{-2(x - 3)}{x^3} \right) \] \[ f''(x) = \frac{-2(x^3) - (-2)(3)(x^2)}{(x^3)^2} \] \[ f''(x) = \frac{-2x^3 + 6x^2}{x^6} \] \[ f''(x) = \frac{-2x^2(x - 3)}{x^6} \] \[ f''(x) = \frac{-2(x - 3)}{x^4} \] The second derivative \( f''(x) = \frac{-2(x - 3)}{x^4} \) is negative for \( x > 3 \) and positive for \( x < 3 \). Therefore, the function is concave down for \( x > 3 \) and concave up for \( x < 3 \).
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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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