AP EAMCET · Maths · Application of Derivatives

Observe the following statements
A : \(f(x)=2 x^3-9 x^2+12 x-3\) is increasing outside the interval \((1,2)\)
\(\mathrm{R}: f^{\prime}(x) < 0\) for \(x \in(1,2)\).
Then, which of the following is true?

A Both A and R are true, and R is not the correct
reason for A
B Both A and R are true, and R is the correct
reason for A
C A is true but R is false
D A is false but R is true

Verified Solution

Answer & Solution

Correct Answer

(A) Both A and R are true, and R is not the correct
reason for A

Step-by-step Solution

Detailed explanation

Statement A For increasing function, \(f^{\prime}(x)>0\) \(\begin{array}{lcrl} & \therefore & & 6\left(x^2-3 x+2\right)>0 \\ \Rightarrow & & 6(x-2)(x-1)>0 \\ \Rightarrow & & x 2 \end{array}\) \(\therefore f(x)\) is increasing outside the interval \((1,2)\), therefore it is true…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Observe the following statementsA : \(f(x)=2 x^3-9 x^2+12 x-3\) is increasing outside the interval \((1,2)\)\(\mathrm{R}: f^{\prime}(x) < 0\) for \(x \in(1,2)\).Then, which of the following is true?

Step-by-step Solution

Observe the following statements
A : \(f(x)=2 x^3-9 x^2+12 x-3\) is increasing outside the interval \((1,2)\)
\(\mathrm{R}: f^{\prime}(x) < 0\) for \(x \in(1,2)\).
Then, which of the following is true?