How do you find concavity when #f(x)= x^(7/3) + x^(4/3)#?
To find concavity, use the second derivative of the function.
With your function:
graph{(28x+4)/(9x^(2/3)) [20.28, 20.27, 10.14, 10.14]}
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To find the concavity of the function f(x) = x^(7/3) + x^(4/3), you need to determine the second derivative and then analyze its sign.

Find the first derivative of f(x) with respect to x. f'(x) = (7/3)x^(4/3) + (4/3)x^(1/3)

Find the second derivative by differentiating f'(x). f''(x) = (28/9)x^(1/3) + (4/9)x^(2/3)

Analyze the sign of the second derivative to determine concavity.
 If f''(x) > 0 for all x in the domain, the function is concave up.
 If f''(x) < 0 for all x in the domain, the function is concave down.
 If f''(x) changes sign at a point, there's a point of inflection.
So, to determine concavity:
 If x > 0, f''(x) > 0, so the function is concave up.
 If x < 0, f''(x) < 0, so the function is concave down.
 There's a point of inflection at x = 0 where the concavity changes.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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