To evaluate the function [tex]\( f(x) = 2x^3 - 3x^2 + 7 \)[/tex] at the specified points, follow these steps:
### 1. Calculate [tex]\( f(-1) \)[/tex]:
Substitute [tex]\( x = -1 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-1) = 2(-1)^3 - 3(-1)^2 + 7 \][/tex]
Now evaluate this expression:
- [tex]\((-1)^3 = -1\)[/tex]
- [tex]\((-1)^2 = 1\)[/tex]
- [tex]\(2 \times (-1) = -2\)[/tex]
- [tex]\(3 \times 1 = 3\)[/tex]
Thus:
[tex]\[ f(-1) = 2(-1) - 3(1) + 7 \][/tex]
[tex]\[ f(-1) = -2 - 3 + 7 \][/tex]
[tex]\[ f(-1) = 2 \][/tex]
So, [tex]\( f(-1) = 2 \)[/tex].
### 2. Calculate [tex]\( f(1) \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(1) = 2(1)^3 - 3(1)^2 + 7 \][/tex]
Now evaluate this expression:
- [tex]\((1)^3 = 1\)[/tex]
- [tex]\((1)^2 = 1\)[/tex]
- [tex]\(2 \times 1 = 2\)[/tex]
- [tex]\(3 \times 1 = 3\)[/tex]
Thus:
[tex]\[ f(1) = 2(1) - 3(1) + 7 \][/tex]
[tex]\[ f(1) = 2 - 3 + 7 \][/tex]
[tex]\[ f(1) = 6 \][/tex]
So, [tex]\( f(1) = 6 \)[/tex].
### 3. Calculate [tex]\( f(2) \)[/tex]:
Substitute [tex]\( x = 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(2) = 2(2)^3 - 3(2)^2 + 7 \][/tex]
Now evaluate this expression:
- [tex]\((2)^3 = 8\)[/tex]
- [tex]\((2)^2 = 4\)[/tex]
- [tex]\(2 \times 8 = 16\)[/tex]
- [tex]\(3 \times 4 = 12\)[/tex]
Thus:
[tex]\[ f(2) = 2(8) - 3(4) + 7 \][/tex]
[tex]\[ f(2) = 16 - 12 + 7 \][/tex]
[tex]\[ f(2) = 11 \][/tex]
So, [tex]\( f(2) = 11 \)[/tex].
### Summary:
[tex]\[
\begin{array}{l}
f(-1) = 2 \\
f(1) = 6 \\
f(2) = 11
\end{array}
\][/tex]
