Edexcel IAL January 2024 P4 (WMA14/01) Q8 Proof by Contradiction
Use proof by contradiction to prove that the curve with equation
\( y = 2x + x^3 + \cos x \)
has no stationary points.
(4)
Solution to Question: Proof by Contradiction
Key Concepts Used:
- Proof by Contradiction: Assume the curve does have stationary points.
- Definition of Stationary Points: \( \frac{dy}{dx} = 0 \).
- Range of Trigonometric Functions: Using the property \( -1 \leq \sin x \leq 1 \) to demonstrate the equation is impossible.
Step-by-Step Solution:
- Form the Assumption
Assume the curve \( y = 2x + x^3 + \cos x \) has stationary points. This means that for some value of \( x \), the gradient \( \frac{dy}{dx} \) is equal to zero.
- Find the Derivative
Differentiate the equation with respect to \( x \):
\( \frac{dy}{dx} = 2 + 3x^2 – \sin x \) - Set to Zero
Set the derivative equal to zero to find stationary points:
\( 2 + 3x^2 – \sin x = 0 \) - Identify the Contradiction
Rearrange the equation to isolate \( \sin x \),
\( \sin x = 2 + 3x^2 \)Consider the possible range of values for the right-hand side, \( 2 + 3x^2 \). Since \( x^2 \geq 0 \), it follows that \( 3x^2 \geq 0 \). Therefore, \( 2 + 3x^2 \geq 2 \). The range of the sine function is \( -1 \leq \sin x \leq 1 \). The equation requires \( \sin x \) to be greater than or equal to 2, which is impossible.
- Conclusion
The assumption leads to an impossible equation.
Final Answer:
Since the assumption that \( \frac{dy}{dx} = 0 \) leads to the contradiction \( \sin x \geq 2 \), the assumption is incorrect. Therefore, the curve \( y = 2x + x^3 + \cos x \) has no stationary points.