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

Answer 1

#f(x)# is concave on (-oo,3), and convex on #(3,oo)#.

To find where a function is concave or convex, find where #f''(x)# is positive or negative (respectively).
#f(x)=(x-2)(x-4)(x-3)#
#f'(x)=(1)(x-4)(x-3)+(x-2)(1)(x-3)+(x-2)(x-4)(1)# #f'(x)=x^2-7x-12+x^2-5x+6+x^2-6x+8# #f'(x)=3x^2-18x+2#
#f''(x)=6x-18# #f''(x)=6(x-3)#
#f''(x)# is negative from #(-oo,3)#, and positive from #(3,oo)#
Therefore, #f(x)# is concave on (-oo,3), and convex on #(3,oo)#.
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Answer 2

To determine the concavity or convexity of the function ( f(x) = (x - 2)(x - 4)(x - 3) ), we need to find its second derivative and then analyze its sign.

First, find the first derivative: [ f'(x) = (x - 4)(x - 3) + (x - 2)(x - 3) + (x - 2)(x - 4) ]

Now, find the second derivative: [ f''(x) = 2(x - 3) + 2(x - 2) + 2(x - 4) ]

[ f''(x) = 2x - 6 + 2x - 4 + 2x - 8 ]

[ f''(x) = 6x - 18 ]

To determine the concavity or convexity, analyze the sign of the second derivative:

  • If ( f''(x) > 0 ) for a certain range, the function is concave up (convex).
  • If ( f''(x) < 0 ) for a certain range, the function is concave down (concave).

For ( f''(x) = 6x - 18 ), it is concave up (convex) when ( x > 3 ) and concave down (concave) when ( x < 3 ).

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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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