Let g: -2, 2→ℝ and f: -2, 2→ℝ are two functions defined as gx=-1, if -2≤x<0x2-1,if 0≤x≤2 and f(x)=|g(x)|+g(|x|)+2. In the interval (-2, 2), f is not differentiable at x=

TS EAMCET · Maths · Continuity and Differentiability

Let $g : [- 2, 2] \to ℝ$ and $f : [- 2, 2] \to ℝ$ are two functions defined as
$g (x) = \{\begin{matrix} - 1, & if - 2 \leq x < 0 \\ x^{2} - 1, & if 0 \leq x \leq 2 \end{matrix}$ and $f (x) = | g (x) | + g (| x |) + 2 .$ In the interval $(- 2, 2), f$ is not differentiable at $x =$

A $0$
B $1$
C $\frac{1}{2}$
D $- 1$

Verified Solution

Answer & Solution

Correct Answer

(B) $1$

Step-by-step Solution

Detailed explanation

gx=-1-2≤x≤0x2-10≤x≤2 =1-2≤x≤01-x20≤x<1x2-11≤x≤2 And gx=-1-2≤x<0x2-10≤x≤2 So, gx=x2-1, -2≤x≤2 Hence, fx=x2-2≤x<000≤x<12x2-11≤x≤2…

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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Let g: -2, 2→ℝ and f: -2, 2→ℝ are two functions defined asgx=-1, if -2≤x<0x2-1,if 0≤x≤2 and f(x)=|g(x)|+g(|x|)+2. In the interval (-2, 2), f is not differentiable at x=

Step-by-step Solution

Let $g : [- 2, 2] \to ℝ$ and $f : [- 2, 2] \to ℝ$ are two functions defined as
$g (x) = \{\begin{matrix} - 1, & if - 2 \leq x < 0 \\ x^{2} - 1, & if 0 \leq x \leq 2 \end{matrix}$ and $f (x) = | g (x) | + g (| x |) + 2 .$ In the interval $(- 2, 2), f$ is not differentiable at $x =$