The function f is defined as $f(x) = -x^3 + 4x^2 + x - 2$, where $x \in \mathbb{R}$ - Leaving Cert Mathematics - Question 3 - 2019
Question 3
The function f is defined as $f(x) = -x^3 + 4x^2 + x - 2$, where $x \in \mathbb{R}$.
(a) (i) Complete the table below for the values of f in the domain $-1 \leq x ... show full transcript
Worked Solution & Example Answer:The function f is defined as $f(x) = -x^3 + 4x^2 + x - 2$, where $x \in \mathbb{R}$ - Leaving Cert Mathematics - Question 3 - 2019
Step 1
Complete the table and draw the graph
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Answer
To complete the table, we evaluate f(x) at each given x in the domain:
For x=−1:
f(−1)=−(−1)3+4(−1)2+(−1)−2=−(−1)+4−1−2=2
For x=0:
f(0)=−03+4(0)2+0−2=−2
For x=1:
f(1)=−(1)3+4(1)2+1−2=−1+4+1−2=2
For x=2:
f(2)=−(2)3+4(2)2+2−2=−8+16+2−2=8
For x=3:
f(3)=−(3)3+4(3)2+3−2=−27+36+3−2=10
For x=4:
f(4)=−(4)3+4(4)2+4−2=−64+64+4−2=2
Thus the completed table is:
x
f(x)
-1
2
0
-2
1
2
2
8
3
10
4
2
Next, plotting these points on a graph will show the shape of the function. The graph should show smooth transitions between the points.
Step 2
Use your graph to estimate the two roots of f(x)
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Answer
From the graph, estimate the roots of f(x) by identifying the x-intercepts, where f(x)=0. The estimated roots in the domain −1≤x≤4 are approximately:
Root 1: x≈−0.7
Root 2: x≈0.7
Step 3
Find the value of x for which f''(x) = 0
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Answer
To find f′′(x), we first differentiate f(x):
f′(x)=−3x2+8x+1
Now, differentiating again gives us:
f′′(x)=−6x+8
Setting f′′(x)=0 results in:
−6x+8=0
Solving for x:
−6x=−8 x=34
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