For what values of x is #f(x)= -x^3+3x^2+2x-12 # concave or convex?

Answer 1

As viewed from O, concave #( y' uarr)# for #x < 1# and concave #(y' darr)# for# x >1. (1, -8)# is the point of inflexion (POI), at which the tangent crosses the curve , for reversing rotation.

#y=f(x)=-x^3+3x^2+2x-12#
#y'=-3x^2+6x+2 =0#, at the turning points#x =1+-sqrt(5/3)=-0.291 and 2.291#, nearly

y''--6(x-1)=0#, at x =1.

y'''=-6 #ne 0#
So, x =1 gives the point of inflexion (POI) #(1. -8)#.

Here, the tangent crosses the curve, reversing rotation, from

anticlockwise to clockwise.

The second graph, the zooming is to see #POI (1. -8)# in #Q_4#,
at the level #y = - 8#.

graph{-x^3+3x^2+2x-12 [-29.95, 29.95, -14.97, 14.98]}

graph{(-x^3+3x^2+2x-12-y)(y+8)=0 [-2, 2, -20, 20]} .

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Answer 2
To determine where the function f(x) = -x^3 + 3x^2 + 2x - 12 is concave or convex, we need to find its second derivative, f''(x), and then analyze its sign. f''(x) = -6x + 6 To find the critical points, set f''(x) = 0 and solve for x: -6x + 6 = 0 x = 1 Now, we can analyze the sign of f''(x) in intervals around the critical point: For x < 1: -6x + 6 < 0, hence f''(x) < 0, indicating concavity. For x > 1: -6x + 6 > 0, hence f''(x) > 0, indicating convexity. Therefore, the function f(x) is concave for x < 1 and convex for x > 1.
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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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