A reliability engineer is modeling the probability p p\,p of a component failing under extreme stress, where the probability is constrained by the open interval 0.2<p<0.30.2 < p < 0.30.2<p<0.3.
The engineer suggests that 0.3 is the maximum possible value for p p\,p within this range. Explain the error in this claim.
To demonstrate that no maximum value exists in this interval, the engineer proposes the following proof by contradiction:
Step 1: Assume there exists a maximum value M M\,M within the interval 0.2<p<0.30.2 < p < 0.30.2<p<0.3.
Step 2: Let q=M+0.32\displaystyle q = \frac{M + 0.3}{2}q=2M+0.3.
Step 3: M<q<0.3M < q < 0.3M<q<0.3, which is a contradiction.
Step 4: Therefore, no maximum value exists in the interval 0.2<p<0.30.2 < p < 0.30.2<p<0.3.
Explain the contradiction stated in Step 3.
Use a similar proof by contradiction to prove that no minimum value exists in the interval 0.2<p<0.30.2 < p < 0.30.2<p<0.3.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.