For what values of x is #f(x)=(3x-2)(4x+2) (x+3)# concave or convex?

Answer 1

for #x< -17/68# convexity and #x > -17/68# concavity

For a twice continuous function like the one proposed, the concavity or convexity is determined by the second derivative sign. #d^2/(dx^2)f(x)=72x+68#, If #d^2/(dx^2)f(x) < 0# the curvature is considered as convex because the area region contained below is a convex set. If #d^2/(dx^2)f(x) > 0# is concave. Solving for #d^2/(dx^2)f(x) = 0# we get #x =-17/68 # so for #x<-17/68# we have convexity and for #x>-17/68# concavity.
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Answer 2

To determine the concavity of the function ( f(x) = (3x - 2)(4x + 2)(x + 3) ), we need to find its second derivative and then analyze its sign. The function will be concave up (convex) where the second derivative is positive and concave down where the second derivative is negative.

First, let's find the first derivative of ( f(x) ) using the product rule: [ f'(x) = (3x - 2)(4x + 2) + (3)(4x + 2)(x + 3) + (3x - 2)(x + 3) ]

Now, let's find the second derivative of ( f(x) ) using the product rule and the sum rule: [ f''(x) = (3)(4x + 2) + (3x - 2)(4) + (3)(x + 3) + (3x - 2) + (3x - 2) ]

Simplify ( f''(x) ): [ f''(x) = 12 + 12x - 8 + 3 + 3x + 9 + 3x - 2 + 3x - 2 ] [ f''(x) = 18x + 12 ]

Now, we analyze the sign of ( f''(x) ) to determine concavity:

  • ( f''(x) > 0 ) (positive) implies the function is concave up (convex).
  • ( f''(x) < 0 ) (negative) implies the function is concave down.

Since ( f''(x) = 18x + 12 ) is a linear function with a positive coefficient for ( x ), it is positive for all ( x ). Therefore, the function ( f(x) ) is concave up (convex) for all real 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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