For what values of x is #f(x)=3x^3-7x^2-5x+9# concave or convex?

Answer 1

#f# is concave (concave down) on #(-oo,7/9)# and is convex (concave up) on #(7/9,oo)#.

The convexity and concavity of the function #f# can be determined by looking at the sign of the second derivative:

To find the function's second derivative, use the power rule.

#f(x)=3x^3-7x^2-5x+9#
#f'(x)=9x^2-14x-5#
#f''(x)=18x-14#
So, the convexity and concavity are determined by the sign of #f''(x)=18x-14#.
The second derivative equals #0# when #18x-14=0#, which is at #x=7/9#.
When #x>7/9#, #f''(x)>0#, so #f(x)# is convex on #(7/9,oo)#.
When #x<7/9#, #f''(x)<0#, so #f(x)# is concave on #(-oo,7/9)#.
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Answer 2
To determine the intervals where the function \(f(x) = 3x^3 - 7x^2 - 5x + 9\) is concave or convex, we need to examine its second derivative, because the sign of the second derivative tells us about the concavity: 1. **First derivative \(f'(x)\):** \[f'(x) = \frac{d}{dx}(3x^3 - 7x^2 - 5x + 9) = 9x^2 - 14x - 5.\] 2. **Second derivative \(f''(x)\):** \[f''(x) = \frac{d}{dx}(9x^2 - 14x - 5) = 18x - 14.\] Set \(f''(x)\) equal to 0 to find critical points: \[18x - 14 = 0 \implies x = \frac{14}{18} = \frac{7}{9}.\] - **For \(x < \frac{7}{9}\):** Choose a test value, say \(x = 0\), \[f''(0) = 18(0) - 14 = -14,\] which is negative. Thus, \(f(x)\) is concave down on this interval. - **For \(x > \frac{7}{9}\):** Choose a test value, say \(x = 1\), \[f''(1) = 18(1) - 14 = 4,\] which is positive. Thus, \(f(x)\) is concave up on this interval. Therefore, the function \(f(x) = 3x^3 - 7x^2 - 5x + 9\) is concave down for \(x < \frac{7}{9}\) and concave up for \(x > \frac{7}{9}\).
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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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