the questions in level 1 are introductory problems. the hints contain links to videos covering this…

the questions in level 1 are introductory problems. the hints contain links to videos covering this content.\nquestion 1 (1 point)\nsuppose ( f(x)=8x^{3}+12x + 2 ) and ( f(1)=-4 ). then ( f(-1) ) equals\n. (enter a number for your answer.)\nview hint for question 1

the questions in level 1 are introductory problems. the hints contain links to videos covering this content.\nquestion 1 (1 point)\nsuppose ( f(x)=8x^{3}+12x + 2 ) and ( f(1)=-4 ). then ( f(-1) ) equals\n. (enter a number for your answer.)\nview hint for question 1

Answer

Explanation:

Step1: Integrate (f'(x))

We know that if (f'(x)=8x^{3}+12x + 2), then by the power rule of integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have: [ \begin{align*} f(x)&=\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 \end{align*} ]

Step2: Find the value of (C)

Since (f(1)=-4), substitute (x = 1) into (f(x)=2x^{4}+6x^{2}+2x + C): [ \begin{align*} f(1)&=2\times(1)^{4}+6\times(1)^{2}+2\times(1)+C\ -4&=2 + 6+2+C\ -4&=10 + C\ C&=-14 \end{align*} ] So (f(x)=2x^{4}+6x^{2}+2x-14)

Step3: Calculate (f(-1))

Substitute (x=-1) into (f(x)=2x^{4}+6x^{2}+2x-14): [ \begin{align*} 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 \end{align*} ]

Answer:

(-8)