How do you know where the graph of f(x) is concave up and where it is concave down for #f(x) = x^3 + x#?

Answer 1

The graph of #f# is concave up on intervals on which #f''(x)# is positive and the graph is concave down where #f''(x)# is negative.

So we need to investigate the sign of #f''(x)#.
#f(x) = x^3 + x#
#f'(x) = 3x^2+1#
#f''(x) = 6x#
Clearly, #f''(x) = 6x# is negative for #x <0# and positive for #x>0#.
So the graph of #f# is concave down on #(-oo,0)# and it is concave up on #(0,oo)#
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Answer 2
To determine where the graph of \( f(x) = x^3 + x \) is concave up or concave down, you need to find the second derivative of the function and analyze its sign. 1. Find the first derivative of \( f(x) \): \[ f'(x) = 3x^2 + 1 \] 2. Find the second derivative of \( f(x) \): \[ f''(x) = 6x \] 3. Set \( f''(x) \) equal to zero to find the points of inflection: \[ 6x = 0 \Rightarrow x = 0 \] 4. Analyze the sign of \( f''(x) \) in intervals determined by the points of inflection (if any). - For \( x < 0 \): \( f''(x) < 0 \), so the graph is concave down. - For \( x > 0 \): \( f''(x) > 0 \), so the graph is concave up. Therefore, the graph of \( f(x) = x^3 + x \) is concave down for \( x < 0 \) and concave up for \( x > 0 \).
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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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