If f(x)={ 3 x^2+12 x-1 ;-1 ≤ x ≤ 2 37-x ; 2<x ≤ 3 . then (A) f(x) is increasing in

step 1 So in this question we have f of x is equal to is equal to three x squared plus 12 x minus 1. X

If f(x)={ 3 x^2+12 x-1 ;-1 ≤ x ≤ 2 37-x ; 2<x...

If $f(x)=\left\{\begin{array}{ll}3 x^{2}+12 x-1 ;-1 \leq x \leq 2 \\ 37-x & ; 2<x \leq 3\end{array}\right.$ then (A) $\mathrm{f}(\mathrm{x})$ is increasing in $[-1,2]$ (B) $\mathrm{f}(\mathbf{x})$ is continuous in $[-1,3]$ (C) $f^{\prime}(2)$ does not exist (D) $f(x)$ has the maximum value at $x=2$

If $f(x)=\left\{\begin{array}{ll}3 x^{2}+12 x-1 ;-1 \leq x \leq 2 \\ 37-x & ; 2<x \leq 3\end{array}\right.$ then
(A) $\mathrm{f}(\mathrm{x})$ is increasing in $[-1,2]$
(B) $\mathrm{f}(\mathbf{x})$ is continuous in $[-1,3]$
(C) $f^{\prime}(2)$ does not exist
(D) $f(x)$ has the maximum value at $x=2$

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by different expressions over different parts of their domain. Understanding piecewise functions involves analyzing each segment separately, especially at the points where the definition changes, to evaluate properties such as continuity and differentiability.

Continuity

Continuity of a function means that the function does not have any breaks, jumps, or holes in its domain. For piecewise functions, it is crucial to check that the function values match at the boundaries where the pieces meet to ensure there are no discontinuities.

Differentiability

Differentiability refers to the existence of a derivative at each point in the function's domain. In the context of piecewise functions, even if each individual piece is differentiable, the function as a whole might fail to be differentiable at the junction points if the left-hand and right-hand derivatives do not agree.

Monotonicity

Monotonicity, particularly increasing or decreasing behavior, is determined by the sign of the derivative over an interval. If the derivative is positive throughout a specific interval, the function is increasing on that interval.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a continuous function on a closed interval will attain both a maximum and a minimum value. This concept is used to analyze where the extreme values occur, which, in piecewise functions, may include endpoints or points where the function's behavior changes.

Please give Ace some feedback