The curve C with equation y = f(x) passes through the point (5, 65). Given that f'(x) = 6x^2 - 10x - 12, a) use integration to find f(x). b) Hence show that f(x) = (2x + 3)(x - 4). c) In the space provided on page 17, sketch C, showing the coordinates of the points where C crosses the x-axis.

A-Level Edexcel Maths Pure Simultaneous Equations: The curve C with equation y = f(

The curve C with equation y = f(x) passes through the point (5, 65) - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 1

Question 10

The curve C with equation y = f(x) passes through the point (5, 65). Given that f'(x) = 6x^2 - 10x - 12, a) use integration to find f(x). b) Hence show that f(x) ... show full transcript

Worked Solution & Example Answer:The curve C with equation y = f(x) passes through the point (5, 65) - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 1

Step 1

a) use integration to find f(x).

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Answer

To find f(x), we need to integrate f'(x):

$f(x) = \int (6x^2 - 10x - 12) \, dx$

Calculating the integral:

$f(x) = 2x^3 - 5x^2 - 12x + C$

Next, we use the point (5, 65) to determine the constant C:

$65 = 2(5)^3 - 5(5)^2 - 12(5) + C$

Simplifying this:

$65 = 2(125) - 5(25) - 60 + C$ $65 = 250 - 125 - 60 + C$ $C = 65 - 65$ $C = 0$

Thus, we have:

$f(x) = 2x^3 - 5x^2 - 12x$

Step 2

b) Hence show that f(x) = (2x + 3)(x - 4).

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Answer

To factor f(x):

$f(x) = 2x^3 - 5x^2 - 12x$

We can factor out 2:

$f(x) = 2(x^3 - \frac{5}{2}x^2 - 6x)$

To factor further, we can use the Rational Root Theorem or synthetic division to find that (x - 4) is a root.

By polynomial division or using the quadratic formula, we can rewrite part of the expression and find:

$f(x) = 2(x + 3)(x - 4).$

Step 3

c) In the space provided on page 17, sketch C, showing the coordinates of the points where C crosses the x-axis.

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Answer

To find where C crosses the x-axis, set f(x) = 0:

$2(x + 3)(x - 4) = 0$

This gives the solutions:

$x = -3, \, x = 4$

Thus, the coordinates where C crosses the x-axis are (-3, 0) and (4, 0). When sketching C, ensure to include these points and note that the curve has local maxima and minima as indicated.

The curve C with equation y = f(x) passes through the point (5, 65) - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 1

Worked Solution & Example Answer:The curve C with equation y = f(x) passes through the point (5, 65) - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 1

a) use integration to find f(x).

b) Hence show that f(x) = (2x + 3)(x - 4).

c) In the space provided on page 17, sketch C, showing the coordinates of the points where C crosses the x-axis.

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