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

MATH102 December 2010

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Question 01 (b)
For this short-answer question, only the answers (placed in the boxes) will be marked. On what interval(s) is the graph of y = 2x4 - 4x3 - 9x2 - 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

Found a typo? Is this solution unclear? Let us know here. 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 $\displaystyle f(x)=2x^{4}-4x^{3}-9x^{2}-x+3$ Its derivative is $\displaystyle f'(x)=8x^{3}-12x^{2}-18x-1$ and its second derivative is $\displaystyle f''(x)=24x^{2}-24x-18=6(4x^{2}-4x-3)$ We need to know the sign of the second derivative, so we find the roots of the polynomial 4x2 - 4x -3. Its discriminant is Δ = 16 - 4*4*(-3) = 64 and so its roots are (4 + 8)/8 = 3/2 and (4-8)/8 = -1/2. Since y = 4x2 - 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 = 2x4 - 4x3 - 9x2 - x + 3 will be concave down on that same interval.

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