Assertion (A) and Reason (R) (A) Both A and R are true and R is the correct explanation of A. (B

step 1 So in this question we have f of x is equal to minus times of x squared plus 4x plus 1 plus sine

Assertion (A) and Reason (R) (A) Both A and R are true and R...

Assertion (A) and Reason (R) (A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$. (B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$. (C) $\mathrm{A}$ is true, $\mathrm{R}$ is false. (D) $\mathrm{A}$ is false, $\mathrm{R}$ is true. Assertion $(\mathrm{A}):$ The least value of the function $f(x)=-x^{2}$ $+4 x+1+\sin ^{-1}\left(\frac{x}{2}\right)$ on the interval $[-1,1]$ is $-4-\frac{\pi}{6}$ Reason $(\mathrm{R}):$ The least value of $\mathrm{f}(\mathrm{x})$ in $[-1,1]=\min (\mathrm{f}(-1)$ $f(1)\}=\min \left\{-4-4 \frac{\pi}{6}, 4+\frac{\pi}{6}\right\}=-4-\frac{\pi}{6}$

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathrm{A}):$ The least value of the function $f(x)=-x^{2}$ $+4 x+1+\sin ^{-1}\left(\frac{x}{2}\right)$ on the interval $[-1,1]$ is $-4-\frac{\pi}{6}$
Reason $(\mathrm{R}):$ The least value of $\mathrm{f}(\mathrm{x})$ in $[-1,1]=\min (\mathrm{f}(-1)$
$f(1)\}=\min \left\{-4-4 \frac{\pi}{6}, 4+\frac{\pi}{6}\right\}=-4-\frac{\pi}{6}$

Key Concepts

Assertion and Reason Questions

This type of question tests the ability to evaluate two related statements independently. One must assess the truthfulness of the assertion and the reason separately, then determine if the reason provides a correct explanation for the assertion. This approach is common in competitive exams and requires clear, logical thinking.

Composite Functions Analysis

Composite function analysis involves understanding functions that are created by combining two or more functions, such as an algebraic function with an inverse trigonometric function. This requires careful consideration of the domains and characteristics of the individual functions, ensuring that the combined function behaves as expected over the interval of interest.

Closed Interval Optimization

This concept pertains to the process of finding maximum or minimum values of a continuous function within a closed interval. According to the Extreme Value Theorem, any continuous function on a closed interval will attain both a maximum and a minimum value. The optimization process typically involves evaluating the function at endpoints as well as at any critical points found within the interval.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the standard trigonometric functions and have specific domains and ranges. They play a critical role in various areas of mathematics, particularly when used in composite functions, since their restricted outputs and inputs must be carefully considered during analysis.

Please give Ace some feedback