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GO Classes Test Series 2023 | Calculus | Test 1 | Question: 15

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Suppose $f$ is twice differentiable with
$$
f^{\prime \prime}(x)=7 x-2, \quad f^{\prime}(-2)=0, \quad \text { and } \quad f(-2)=-2 .
$$
Find $f(0)$.
  1. $-337 / 6$
  2. $-74 / 3$
  3. $23 / 9$
  4. $37 / 4$
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$$
\begin{gathered}
f^{\prime}(x)=\int f^{\prime \prime}(x) d x=\int(7 x-2) d x=\frac{7}{2} x^{2}-2 x+C_{1} \\
0=f^{\prime}(-2)=\frac{7}{2}(-2)^{2}-2(-2)+C_{1}=14+4+C_{1}=18+C_{1} \\
0=\left(8+C_{1}\right. \\
C_{1}=-18 \\
f^{\prime}(x)=\frac{7}{2} \lambda^{2}-2 x-18 \\
f(x)=\int f^{\prime}(x) d x=\int\left(\frac{7}{2} x^{2}-2 x-18\right) d x=\frac{7}{6} x^{3}-x^{2}-18 x+C_{2} \\
-2=f(-2)=\frac{7}{6}(-2)^{3}-(-2)^{2}-18(-2)+C_{2}=-\frac{28}{3}-4+36+C_{2}=\frac{68}{3}+C_{2} \\
-2=\frac{68}{3}+C_{2} \\
C_{2}=-\frac{74}{3}
\end{gathered}
$$
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