Answer :
We are given the function
[tex]$$
f(x)=2x^3-3x^2+7.
$$[/tex]
Our goal is to compute the values for [tex]$f(-1)$[/tex], [tex]$f(1)$[/tex], and [tex]$f(2)$[/tex].
Step 1. Compute [tex]$f(-1)$[/tex]:
[tex]\[
f(-1)=2(-1)^3-3(-1)^2+7.
\][/tex]
Calculate each term:
- [tex]$(-1)^3 = -1$[/tex], so [tex]$2(-1)^3 = 2(-1) = -2$[/tex].
- [tex]$(-1)^2 = 1$[/tex], so [tex]$-3(-1)^2 = -3(1) = -3$[/tex].
- The constant is [tex]$7$[/tex].
Now combine the terms:
[tex]\[
f(-1) = -2 - 3 + 7 = 2.
\][/tex]
Step 2. Compute [tex]$f(1)$[/tex]:
[tex]\[
f(1)=2(1)^3-3(1)^2+7.
\][/tex]
Calculate each term:
- [tex]$(1)^3 = 1$[/tex], so [tex]$2(1)^3 = 2(1) = 2$[/tex].
- [tex]$(1)^2 = 1$[/tex], so [tex]$-3(1)^2 = -3(1) = -3$[/tex].
- The constant remains [tex]$7$[/tex].
Now combine them:
[tex]\[
f(1) = 2 - 3 + 7 = 6.
\][/tex]
Step 3. Compute [tex]$f(2)$[/tex]:
[tex]\[
f(2)=2(2)^3-3(2)^2+7.
\][/tex]
Calculate each term:
- [tex]$(2)^3 = 8$[/tex], so [tex]$2(2)^3 = 2 \times 8 = 16$[/tex].
- [tex]$(2)^2 = 4$[/tex], so [tex]$-3(2)^2 = -3 \times 4 = -12$[/tex].
- Again, the constant is [tex]$7$[/tex].
Combine these values:
[tex]\[
f(2) = 16 - 12 + 7.
\][/tex]
First, [tex]$16 - 12 = 4$[/tex], and then adding [tex]$7$[/tex] gives:
[tex]\[
f(2)= 4+7 = 11.
\][/tex]
Thus, the computed results are:
[tex]\[
f(-1)=2, \quad f(1)=6, \quad \text{and} \quad f(2)=11.
\][/tex]
The final answer for the value of [tex]$f(2)$[/tex] is [tex]$\boxed{11}$[/tex].
[tex]$$
f(x)=2x^3-3x^2+7.
$$[/tex]
Our goal is to compute the values for [tex]$f(-1)$[/tex], [tex]$f(1)$[/tex], and [tex]$f(2)$[/tex].
Step 1. Compute [tex]$f(-1)$[/tex]:
[tex]\[
f(-1)=2(-1)^3-3(-1)^2+7.
\][/tex]
Calculate each term:
- [tex]$(-1)^3 = -1$[/tex], so [tex]$2(-1)^3 = 2(-1) = -2$[/tex].
- [tex]$(-1)^2 = 1$[/tex], so [tex]$-3(-1)^2 = -3(1) = -3$[/tex].
- The constant is [tex]$7$[/tex].
Now combine the terms:
[tex]\[
f(-1) = -2 - 3 + 7 = 2.
\][/tex]
Step 2. Compute [tex]$f(1)$[/tex]:
[tex]\[
f(1)=2(1)^3-3(1)^2+7.
\][/tex]
Calculate each term:
- [tex]$(1)^3 = 1$[/tex], so [tex]$2(1)^3 = 2(1) = 2$[/tex].
- [tex]$(1)^2 = 1$[/tex], so [tex]$-3(1)^2 = -3(1) = -3$[/tex].
- The constant remains [tex]$7$[/tex].
Now combine them:
[tex]\[
f(1) = 2 - 3 + 7 = 6.
\][/tex]
Step 3. Compute [tex]$f(2)$[/tex]:
[tex]\[
f(2)=2(2)^3-3(2)^2+7.
\][/tex]
Calculate each term:
- [tex]$(2)^3 = 8$[/tex], so [tex]$2(2)^3 = 2 \times 8 = 16$[/tex].
- [tex]$(2)^2 = 4$[/tex], so [tex]$-3(2)^2 = -3 \times 4 = -12$[/tex].
- Again, the constant is [tex]$7$[/tex].
Combine these values:
[tex]\[
f(2) = 16 - 12 + 7.
\][/tex]
First, [tex]$16 - 12 = 4$[/tex], and then adding [tex]$7$[/tex] gives:
[tex]\[
f(2)= 4+7 = 11.
\][/tex]
Thus, the computed results are:
[tex]\[
f(-1)=2, \quad f(1)=6, \quad \text{and} \quad f(2)=11.
\][/tex]
The final answer for the value of [tex]$f(2)$[/tex] is [tex]$\boxed{11}$[/tex].
