For what values of x is #f(x)=3+x^2/2# concave or convex?

Answer 1

#f# is convex on #(-oo,+oo)#.

#f# is convex when #f''>0#. #f# is concave when #f''<0#.
First, find #f'(x)#, then #f''(x)#.
#f(x)=3+1/2x^2#
#f'(x)=x#
#f''(x)=1#
Since #f''(x)# is always #>0#, #f# is convex on #(-oo,+oo)#.

graph{3+(x^2)/2 [-14.96, 17.08, -1.67, 14.35]}

The answer makes logical since, since #f# is a parabola that opens up, so it will always be in the convex shape.
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Answer 2

To determine the concavity or convexity of ( f(x) = 3 + \frac{x^2}{2} ), we need to find the second derivative of the function and then analyze its sign.

Given ( f(x) = 3 + \frac{x^2}{2} ), its first derivative is ( f'(x) = x ). Taking the second derivative of ( f(x) ), we get ( f''(x) = 1 ).

Since the second derivative is a constant (1), it is always positive, indicating that the function is convex for all values of ( x ).

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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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