Column- I Column-1I (A) If the maximum and minimum value of...

Column- I Column-1I (A) If the maximum and minimum value of the function (P) 1 $h(y)=y^{3}-6 y^{2}+9 y+1$ on $[0,2]$ are $M$ and $m$ respectively, then $(\mathrm{M}+\mathrm{m})$ is equal to (B) If the maximum and minimum value of the function $f(x)=\tan ^{-1} x-\frac{1}{2} \ell n x$ (Q) 6 on $\left[\frac{1}{\sqrt{3}}, \sqrt{3}\right]$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then $[\mathrm{M}+\mathrm{m}]$ is equal to (where [.] denotes greatest integer function) (C) If the maximum and minimum value of the function (R) $\underline{0}$ $h(y)=\left\{\begin{array}{cc}2 y^{2}+\frac{2}{y^{2}}, & |y| \in[1,2] \\ 1 & |y|<1\end{array}\right.$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then $\mathrm{M}+\mathrm{m}$ is equal to (D) For a given $\mathrm{n} \in \mathrm{N}$, number of real solutions of the (S) $19 / 2$ equation $\min \left(\mathrm{e}^{x}, \mathrm{x}^{2}\right)=\mathrm{n}$ is

Column- I Column-1I
(A) If the maximum and minimum value of the function
(P) 1 $h(y)=y^{3}-6 y^{2}+9 y+1$ on $[0,2]$ are $M$ and $m$ respectively, then $(\mathrm{M}+\mathrm{m})$ is equal to
(B) If the maximum and minimum value of the function $f(x)=\tan ^{-1} x-\frac{1}{2} \ell n x$
(Q) 6 on $\left[\frac{1}{\sqrt{3}}, \sqrt{3}\right]$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then $[\mathrm{M}+\mathrm{m}]$ is equal to (where [.] denotes greatest integer function)
(C) If the maximum and minimum value of the function
(R) $\underline{0}$ $h(y)=\left\{\begin{array}{cc}2 y^{2}+\frac{2}{y^{2}}, & |y| \in[1,2] \\ 1 & |y|<1\end{array}\right.$ are $\mathrm{M}$ and $\mathrm{m}$ respectively, then
$\mathrm{M}+\mathrm{m}$ is equal to
(D) For a given $\mathrm{n} \in \mathrm{N}$, number of real solutions of the
(S) $19 / 2$ equation $\min \left(\mathrm{e}^{x}, \mathrm{x}^{2}\right)=\mathrm{n}$ is

Key Concepts

Equations Involving the Minimum/Maximum Operator

When dealing with equations that incorporate the minimum or maximum of two functions, it is necessary to split the analysis into distinct cases. This involves determining the regions where each function dominates so that the overall equation can be solved correctly over the respective intervals.

Extreme Value Theorem

This fundamental result in calculus asserts that any continuous function defined on a closed interval will achieve both a maximum and minimum value within that interval. It is crucial in optimization problems as it guarantees the existence of extreme values, allowing one to focus on analyzing endpoints and critical points.

Critical Points and Derivative Tests

Determining where the derivative of a function is zero or undefined helps locate potential local maximums or minimums. By computing the derivative and identifying these critical points, one can systematically evaluate candidates for extreme values, which is essential for optimizing functions on a given domain.

Optimization of Piecewise Functions

When a function is defined by different expressions over separate intervals, the analysis must be done piece by piece. This involves checking the behavior and extremes in each defined segment, as well as the junction points where the definition changes, ensuring all potential maxima and minima are considered.

Greatest Integer Function

The greatest integer function, also known as the floor function, assigns to any real number the largest integer that is not greater than the number. Its step-like, non-continuous nature requires careful treatment, especially when it is combined with continuously varying results from optimization problems.

Monotonicity and Behavior of Inverse Trigonometric and Logarithmic Functions

Understanding the properties such as monotonicity, continuity, and differentiability of inverse trigonometric and logarithmic functions is vital. These characteristics are essential when these functions are part of more complex expressions, as they impact how the composite function behaves and how its extrema might be determined.

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