The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2
Question 7
The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$.
(a) Find the exact coordinates of $P$.
(b) The equation $f(x) = 0$ has a root between $x... show full transcript
Worked Solution & Example Answer:The curve with equation $y = f(x) = 3x e^x - 1$ has a turning point $P$ - Edexcel - A-Level Maths Pure - Question 7 - 2009 - Paper 2
Step 1
Find the exact coordinates of P.
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Answer
To find the turning point P, we first need to calculate the first derivative of the function:
f'(x) = rac{d}{dx}(3x e^x - 1) = 3e^x + 3xe^x.
Setting the first derivative equal to zero gives us:
3ex(1+x)=0.
Since ex is never zero, we focus on the term (1+x)=0, leading to x=−1.
To find the corresponding y-coordinate:
f(−1)=3(−1)e−1−1=−e3−1.
Thus, the exact coordinates of P are ig(-1, -\frac{3}{e} - 1\big).
Step 2
Use the iterative formula
with $x_0 = 0.25$ to find, to 4 decimal places, the values of $x_1$, $x_2$, and $x_3$.
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Answer
Using the iterative formula
xn+1=31e−xn,
we start with x0=0.25:
For x1:
x1=31e−0.25≈0.2596.
For x2:
x2=31e−0.2596≈0.2571.
For x3:
x3=31e−0.2571≈0.2578.
Hence, the values are approximately:
x1≈0.2596,
x2≈0.2571,
x3≈0.2578.
Step 3
By choosing a suitable interval, show that a root of $f(x) = 0$ is $x = 0.2576$ correct to 4 decimal places.
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Answer
To demonstrate the existence of a root, we evaluate f(x) at suitable points within the interval (0.2575,0.25765):
Calculate f(0.25765):
f(0.25765)=3(0.25765)e0.25765−1≈−0.000109...
Thus, since f(0.2575)>0 and f(0.25765)<0, by the Intermediate Value Theorem, there is at least one root in the interval (0.2575,0.25765). Therefore, we conclude that a root of f(x)=0 is approximately x=0.2576, accurate to 4 decimal places.
