Question 296583
find the largest open interval at which function is concave up or concave down and find the location of any points of inflection. 
f(x)= x^4+8x^3-30x^2+24x-3 
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f'(x) = 4x^3
 + 24x^2 - 60x + 24
Solve f'(x) = 0
x = -7.975.. or x = 0.5154 or x = 1.4598
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f"(x) = 12x^2 + 48x - 60
Find f"(-7.975) > 0 so f(x) is concave up in that region
Find f"(0.5154) < 0 so f(x) is concave down in that region
Find f"(1.4598) > 0 so f(x) is concave up in that region
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Solve f"(x)= 0
x = x = -0.5 or x = 1
This locates points of inflection.
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Concave up: (-inf,-0.5)U(1,+inf)
Concave down: (-0.5,1)
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Cheers,
Stan H.