AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. Lef f be the function definca by f(a)=int_(0)^(1)(2z^(2)-15z^(2):}. ini) dt. On which of the following inten als is the graph of y=f(x) concavc down? (A) (-oo,0) only (B) (rarr oo,2) (C) (0,oo) (D) (2,3) only (E) (3,oo) only

Question

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. Lef  f be the function definca by  f(a)=int_(0)^(1)(2z^(2)-15z^(2):}. ini) dt. On which of the following inten als is the graph of  y=f(x) concavc down? (A)  (-oo,0) only (B)  (rarr oo,2) (C)  (0,oo) (D)  (2,3) only (E)  (3,oo) only

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Identify Function:  Identify the function inside the integral. Reasoning: The function given inside the integral is $2z^2 - 15z + 2$. Calculation: No calculation needed here.Calculate Integral:  Calculate the integral of the function from 0 to 1. Reasoning: To find $f(x)$, we need to integrate the quadratic function. Calculation: $ \int_0^1 (2z^2 - 15z + 2) \, dz = \left[\frac{2}{3}z^3 - \frac{15}{2}z^2 + 2z\right]_0^1 = \left(\frac{2}{3} - \frac{15}{2} + 2\right) - (0) = \frac{2}{3} - 7.5 + 2 = -2.83 $.Determine Second Derivative:  Determine the second derivative of the function. Reasoning: Concavity is determined by the second derivative of the function. Calculation: Since $f(x)$ is a constant (as the integral of a function over a fixed interval results in a constant), $f''(x) = 0$.Analyze Concavity:  Analyze the second derivative to find concavity. Reasoning: A function is concave down where its second derivative is less than 0. Calculation: Since $f''(x) = 0$ everywhere, there are no intervals where the function is concave down.

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