How do you determine whether the function #f(x) = x^2e^x# is concave up or concave down and its intervals?
You have to calculate inflection point(s), which mean find zeros of the second derivative.
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To determine the concavity of the function ( f(x) = x^2e^x ), we need to find its second derivative and then analyze its sign.
First, find the first derivative of ( f(x) ) using the product rule:
[ f'(x) = (2x \cdot e^x) + (x^2 \cdot e^x) ]
Now, differentiate ( f'(x) ) with respect to ( x ) again to find the second derivative:
[ f''(x) = (2 \cdot e^x + 2x \cdot e^x) + (2x \cdot e^x + x^2 \cdot e^x) ]
[ f''(x) = (2 + 2x) \cdot e^x + (2x + x^2) \cdot e^x ]
[ f''(x) = (2 + 2x + 2x + x^2) \cdot e^x ]
[ f''(x) = (x^2 + 4x + 2) \cdot e^x ]
To determine the concavity of the function, analyze the sign of the second derivative. If ( f''(x) > 0 ), the function is concave up. If ( f''(x) < 0 ), the function is concave down.
The second derivative ( f''(x) = (x^2 + 4x + 2) \cdot e^x ) is always positive for all real values of ( x ). Therefore, the function ( f(x) = x^2e^x ) is concave up for all real values of ( x ). There are no intervals where it is concave down.
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To determine the concavity of the function (f(x) = x^2e^x) and its intervals, you need to find the second derivative of the function and then analyze its sign.
- Find the first derivative of (f(x)) using the product rule: (f'(x) = 2xe^x + x^2e^x).
- Find the second derivative of (f(x)) using the product rule again: (f''(x) = (2e^x + 2xe^x) + (2xe^x + x^2e^x) = (2 + 2x)e^x + 2xe^x = (2 + 4x)e^x).
- To determine concavity, set (f''(x)) equal to zero and solve for (x): ((2 + 4x)e^x = 0). Since (e^x) is always positive, the only way for the expression to be zero is if (2 + 4x = 0). Solving for (x), we get (x = -\frac{1}{2}).
- Choose test points on either side of (x = -\frac{1}{2}) to determine the concavity intervals.
- For (x < -\frac{1}{2}), pick (x = -1). (f''(-1) = (2 + 4(-1))e^{-1} = (-2)e^{-1} < 0), so the function is concave down on (-\infty < x < -\frac{1}{2}).
- For (x > -\frac{1}{2}), pick (x = 0). (f''(0) = (2 + 4(0))e^{0} = 2 > 0), so the function is concave up on (-\frac{1}{2} < x < \infty).
Therefore, the function (f(x) = x^2e^x) is concave down on (-\infty < x < -\frac{1}{2}) and concave up on (-\frac{1}{2} < x < \infty).
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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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