Question

suppose $f(x)=8x^{3}+12x + 2$ and $f(1)=-4$. then $f(-1)$ equals (enter a number for your answer.)

Explanation:

Step1: Integrate \(f^{\prime}(x)\)

Integrate \(f^{\prime}(x)=8x^{3}+12x + 2\) term - by - term.
Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have:
\(\int(8x^{3}+12x + 2)dx=8\int x^{3}dx+12\int xdx+2\int dx\)
\(=8\times\frac{x^{4}}{4}+12\times\frac{x^{2}}{2}+2x+C\)
\(=2x^{4}+6x^{2}+2x + C\). So \(f(x)=2x^{4}+6x^{2}+2x + C\)

Step2: Find the value of \(C\)

Since \(f(1)=-4\), substitute \(x = 1\) into \(f(x)\):
\(f(1)=2\times1^{4}+6\times1^{2}+2\times1+C\)
\(2 + 6+2+C=-4\)
\(10 + C=-4\), then \(C=-14\)

Step3: Find \(f(-1)\)

Substitute \(x=-1\) and \(C = - 14\) into \(f(x)=2x^{4}+6x^{2}+2x-14\)
\(f(-1)=2\times(-1)^{4}+6\times(-1)^{2}+2\times(-1)-14\)
\(=2\times1+6\times1-2 - 14\)
\(=2 + 6-2-14\)
\(=-8\)

Answer:

\(-8\)