Science:Math Exam Resources/Courses/MATH102/December 2010/Question 01 (b)
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Question 01 (b) 

For this shortanswer question, only the answers (placed in the boxes) will be marked. On what interval(s) is the graph of y = 2x^{4}  4x^{3}  9x^{2}  x + 3 concave down? 
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Hint 

What is the link between the concavity of a graph and its second derivative? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The graph of a function is concave down where its second derivative is negative. We are given the function Its derivative is and its second derivative is We need to know the sign of the second derivative, so we find the roots of the polynomial 4x^{2}  4x 3. Its discriminant is Δ = 16  4*4*(3) = 64 and so its roots are (4 + 8)/8 = 3/2 and (48)/8 = 1/2. Since y = 4x^{2}  4x 3 is a concave up parabola, we know it will be negative on the interval (1/2, 3/2) and positive elsewhere and hence the curve y = 2x^{4}  4x^{3}  9x^{2}  x + 3 will be concave down on that same interval. 