f(x) = x^4 - 4x - 8.
(a) Show that there is a root of f(x) = 0 in the interval [-2, -1].
(b) Find the coordinates of the turning point on the graph of y = f(x).
(... show full transcript
Worked Solution & Example Answer:f(x) = x^4 - 4x - 8 - Edexcel - A-Level Maths Pure - Question 8 - 2007 - Paper 6
Step 1
Show that there is a root of f(x) = 0 in the interval [-2, -1]
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Answer
To find a root of the equation in the interval [-2, -1], we evaluate f at the endpoints:
Calculate f(-2):
f(−2)=(−2)4−4(−2)−8=16+8−8=16>0
Calculate f(-1):
f(−1)=(−1)4−4(−1)−8=1+4−8=−3<0
Since f(-2) > 0 and f(-1) < 0, by the Intermediate Value Theorem, there must be a root in the interval [-2, -1] because of the sign change.
Step 2
Find the coordinates of the turning point on the graph of y = f(x)
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Answer
To find the turning point, we first find the derivative of f(x):
f′(x)=4x3−4
Setting this to zero to find critical points:
ightarrow x^3 = 1
ightarrow x = 1$$
Next, we calculate the corresponding y-coordinate:
$$f(1) = (1)^4 - 4(1) - 8 = 1 - 4 - 8 = -11$$
Thus, the coordinates of the turning point are (1, -11).
Step 3
Given that f(x) = (x - 2)(x^2 + ax^2 + bx + c), find the values of the constants, a, b and c
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Answer
Expanding the expression:
f(x)=(x−2)(x2+ax2+bx+c)=x3(1+a)+x2(b−2)+c(−2)
Comparing coefficients with the original polynomial, we need:
The coefficient of x^3: 1+a=0ightarrowa=−1
The coefficient of x: b−2=−4ightarrowb=−2
The constant term: −2c=−8ightarrowc=4
Thus, the values are: a=−1,b=−2,c=4.
Step 4
In the space provided on page 21, sketch the graph of y = f(x)
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To sketch the graph:
Mark the turning point (1, -11).
Indicate the roots found from part (a) in the interval [-2, -1].
Note the shape of the graph as it will approach infinity as x approaches ±∞, and it has a local maximum and minimum point.
Draw the graph with a smooth curve, ensuring the end behavior is consistent with the leading coefficient being positive.
Step 5
Hence sketch the graph of y = |f(x)|
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For y = |f(x)|:
Reflect the portion of the graph of f(x) that is below the x-axis (i.e., where f(x) < 0) to above the x-axis.
Keep the portion that is above the x-axis unchanged.
Ensure the graph remains smooth without sharp points.
