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Most Important Selected Qs for JEE AdvancedMathematicsContinuity and Differentiability

Let f(x)=x^3-x^2+x+1 and g(x)= array ll . f(t) ; 0 t x, & for 0 x 1 3-x, & for 1 < x 2 array . , then g(x) is:

Options

  1. Acontinuous for x [0,2]- 1
  2. Bcontinuous for x [0,2]
  3. Cderivable for all x [0,2]
  4. Dderivable for all x [0,2]- 1

Correct answer

D. derivable for all x [0,2]- 1

Step-by-step solution

f(x)=x^3-x^2+x+1 f^ (x)=3 x^2-2 x+1>0, f is increasing. g(x)= array cc f(x)=x^3-x^2+x+1, & 0 x 1 3-x, & 1 x 2 array . Clearly g(x) is continuous for all x [0,2] and g^ (x)= array cc 3 x^3-2 x+1, & 0 < x < 1 -1, & 1 < x < 2 array . g^ (1⁻ )=2 and g^ (1⁺ )=-1 g is non-derivable at x=1

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