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

for

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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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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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- What are the points of inflection of #f(x)=(3x^2 + 8x + 5)/(4-x) #?
- Is #f(x)=sinx# concave or convex at #x=-1#?
- How do you find the inflection points of #f(x)=3x^5-5x^4-40x^3+120x^2#?

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