Answer :
We are given the function
[tex]$$
f(x)=2x^4+x^3-49x^2+79x+15.
$$[/tex]
We will evaluate this function at [tex]$x=-1$[/tex], [tex]$x=3$[/tex], and [tex]$x=4$[/tex].
─────────────────────────────
Step 1. Compute [tex]$f(-1)$[/tex]
Substitute [tex]$x=-1$[/tex] into the function:
[tex]$$
f(-1)=2(-1)^4+(-1)^3-49(-1)^2+79(-1)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(-1)^4$[/tex]:
[tex]$$
(-1)^4=1,\quad \text{so}\quad 2(-1)^4=2\cdot1=2.
$$[/tex]
2. Calculate [tex]$(-1)^3$[/tex]:
[tex]$$
(-1)^3=-1.
$$[/tex]
3. Calculate [tex]$(-1)^2$[/tex]:
[tex]$$
(-1)^2=1,\quad \text{so}\quad -49(-1)^2=-49\cdot1=-49.
$$[/tex]
4. Calculate [tex]$79(-1)$[/tex]:
[tex]$$
79(-1)=-79.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(-1)=2 + (-1) - 49 - 79 + 15.
$$[/tex]
Combine them sequentially:
- [tex]$2 - 1 = 1$[/tex]
- [tex]$1 - 49 = -48$[/tex]
- [tex]$-48 - 79 = -127$[/tex]
- [tex]$-127 + 15 = -112$[/tex]
Thus,
[tex]$$
f(-1)=-112.
$$[/tex]
─────────────────────────────
Step 2. Compute [tex]$f(3)$[/tex]
Substitute [tex]$x=3$[/tex] into the function:
[tex]$$
f(3)=2(3)^4+(3)^3-49(3)^2+79(3)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(3)^4$[/tex]:
[tex]$$
3^4=81,\quad \text{so}\quad 2(3)^4=2\cdot81=162.
$$[/tex]
2. Calculate [tex]$(3)^3$[/tex]:
[tex]$$
3^3=27.
$$[/tex]
3. Calculate [tex]$(3)^2$[/tex]:
[tex]$$
3^2=9,\quad \text{so}\quad -49(3)^2=-49\cdot9=-441.
$$[/tex]
4. Calculate [tex]$79(3)$[/tex]:
[tex]$$
79(3)=237.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(3)=162+27-441+237+15.
$$[/tex]
Combine them step by step:
- [tex]$162+27=189$[/tex]
- [tex]$189-441=-252$[/tex]
- [tex]$-252+237=-15$[/tex]
- [tex]$-15+15=0$[/tex]
Thus,
[tex]$$
f(3)=0.
$$[/tex]
─────────────────────────────
Step 3. Compute [tex]$f(4)$[/tex]
Substitute [tex]$x=4$[/tex] into the function:
[tex]$$
f(4)=2(4)^4+(4)^3-49(4)^2+79(4)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(4)^4$[/tex]:
[tex]$$
4^4=256,\quad \text{so}\quad 2(4)^4=2\cdot256=512.
$$[/tex]
2. Calculate [tex]$(4)^3$[/tex]:
[tex]$$
4^3=64.
$$[/tex]
3. Calculate [tex]$(4)^2$[/tex]:
[tex]$$
4^2=16,\quad \text{so}\quad -49(4)^2=-49\cdot16=-784.
$$[/tex]
4. Calculate [tex]$79(4)$[/tex]:
[tex]$$
79(4)=316.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(4)=512+64-784+316+15.
$$[/tex]
Combine them step by step:
- [tex]$512+64=576$[/tex]
- [tex]$576-784=-208$[/tex]
- [tex]$-208+316=108$[/tex]
- [tex]$108+15=123$[/tex]
Thus,
[tex]$$
f(4)=123.
$$[/tex]
─────────────────────────────
Final Answers:
a. [tex]$f(-1)=-112$[/tex]
b. [tex]$f(3)=0$[/tex]
c. [tex]$f(4)=123$[/tex]
[tex]$$
f(x)=2x^4+x^3-49x^2+79x+15.
$$[/tex]
We will evaluate this function at [tex]$x=-1$[/tex], [tex]$x=3$[/tex], and [tex]$x=4$[/tex].
─────────────────────────────
Step 1. Compute [tex]$f(-1)$[/tex]
Substitute [tex]$x=-1$[/tex] into the function:
[tex]$$
f(-1)=2(-1)^4+(-1)^3-49(-1)^2+79(-1)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(-1)^4$[/tex]:
[tex]$$
(-1)^4=1,\quad \text{so}\quad 2(-1)^4=2\cdot1=2.
$$[/tex]
2. Calculate [tex]$(-1)^3$[/tex]:
[tex]$$
(-1)^3=-1.
$$[/tex]
3. Calculate [tex]$(-1)^2$[/tex]:
[tex]$$
(-1)^2=1,\quad \text{so}\quad -49(-1)^2=-49\cdot1=-49.
$$[/tex]
4. Calculate [tex]$79(-1)$[/tex]:
[tex]$$
79(-1)=-79.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(-1)=2 + (-1) - 49 - 79 + 15.
$$[/tex]
Combine them sequentially:
- [tex]$2 - 1 = 1$[/tex]
- [tex]$1 - 49 = -48$[/tex]
- [tex]$-48 - 79 = -127$[/tex]
- [tex]$-127 + 15 = -112$[/tex]
Thus,
[tex]$$
f(-1)=-112.
$$[/tex]
─────────────────────────────
Step 2. Compute [tex]$f(3)$[/tex]
Substitute [tex]$x=3$[/tex] into the function:
[tex]$$
f(3)=2(3)^4+(3)^3-49(3)^2+79(3)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(3)^4$[/tex]:
[tex]$$
3^4=81,\quad \text{so}\quad 2(3)^4=2\cdot81=162.
$$[/tex]
2. Calculate [tex]$(3)^3$[/tex]:
[tex]$$
3^3=27.
$$[/tex]
3. Calculate [tex]$(3)^2$[/tex]:
[tex]$$
3^2=9,\quad \text{so}\quad -49(3)^2=-49\cdot9=-441.
$$[/tex]
4. Calculate [tex]$79(3)$[/tex]:
[tex]$$
79(3)=237.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(3)=162+27-441+237+15.
$$[/tex]
Combine them step by step:
- [tex]$162+27=189$[/tex]
- [tex]$189-441=-252$[/tex]
- [tex]$-252+237=-15$[/tex]
- [tex]$-15+15=0$[/tex]
Thus,
[tex]$$
f(3)=0.
$$[/tex]
─────────────────────────────
Step 3. Compute [tex]$f(4)$[/tex]
Substitute [tex]$x=4$[/tex] into the function:
[tex]$$
f(4)=2(4)^4+(4)^3-49(4)^2+79(4)+15.
$$[/tex]
Now, compute each term:
1. Calculate [tex]$(4)^4$[/tex]:
[tex]$$
4^4=256,\quad \text{so}\quad 2(4)^4=2\cdot256=512.
$$[/tex]
2. Calculate [tex]$(4)^3$[/tex]:
[tex]$$
4^3=64.
$$[/tex]
3. Calculate [tex]$(4)^2$[/tex]:
[tex]$$
4^2=16,\quad \text{so}\quad -49(4)^2=-49\cdot16=-784.
$$[/tex]
4. Calculate [tex]$79(4)$[/tex]:
[tex]$$
79(4)=316.
$$[/tex]
5. The constant term is [tex]$15$[/tex].
Now, add all the terms:
[tex]$$
f(4)=512+64-784+316+15.
$$[/tex]
Combine them step by step:
- [tex]$512+64=576$[/tex]
- [tex]$576-784=-208$[/tex]
- [tex]$-208+316=108$[/tex]
- [tex]$108+15=123$[/tex]
Thus,
[tex]$$
f(4)=123.
$$[/tex]
─────────────────────────────
Final Answers:
a. [tex]$f(-1)=-112$[/tex]
b. [tex]$f(3)=0$[/tex]
c. [tex]$f(4)=123$[/tex]
