TS EAMCET · Maths · Quadratic Equation

Assertion $(A) :$ The maximum value of $- x^{2} + 3 x + 1$ is $\frac{11}{4} .$ Reason $(R) :$ If $a < 0,$ the maximum value of $a x^{2} + b x + c$ exist at $x = \frac{- b}{2 a} .$ The correct option among the following is

A $(A)$ is true, $(R)$ is true and $(R)$ is the correct explanation for $(A)$
B $(A)$ is true, $(R)$ is true but $(R)$ is not the correct explanation for $(A)$
C $(A)$ is true but $(R)$ is false
D $(A)$ is false but $(R)$ is true

Verified Solution

Answer & Solution

Correct Answer

(D) $(A)$ is false but $(R)$ is true

Step-by-step Solution

Detailed explanation

Assertion A: Let y=-x2+3x+1. To find the maximum or minimum value of y differentiate both sides w.r.to x, to get dydx=-2x+3 And to find the maximum or minimum value, put dydx=0 ⇒-2x+3=0 ⇒x=32 and d2ydx2=-2<0. And, we know that at the point of maxima, the second…

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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Assertion A: The maximum value of -x2+3x+1 is 114. Reason R: If a<0, the maximum value of ax2+bx+c exist at x=-b2a. The correct option among the following is

Step-by-step Solution

Assertion $(A) :$ The maximum value of $- x^{2} + 3 x + 1$ is $\frac{11}{4} .$ Reason $(R) :$ If $a < 0,$ the maximum value of $a x^{2} + b x + c$ exist at $x = \frac{- b}{2 a} .$ The correct option among the following is