Class 12 Mathematics: Advanced Continuity and Differentiability

For what value of 'k' is the function f(x) = { (sin(k+1)x + sin x) / x , if x < 0 ; c , if x = 0 ; (sqrt(x+bx^2) - sqrt(x)) / (bx^(3/2)) , if x > 0 } continuous at x = 0?

If f(x) = |x-1| + |x-2| + |x-3|, then f(x) is not differentiable at:

Let f(x) = { e^(1/x) / (1 + e^(1/x)) , if x ≠ 0 ; 0 , if x = 0 }. Which of the following statements is true regarding the continuity of f(x) at x = 0?

If y = cot⁻¹( (√(1+sin x) + √(1-sin x)) / (√(1+sin x) - √(1-sin x)) ), then dy/dx is equal to:

The derivative of tan⁻¹((√(1+x²) - 1)/x) with respect to tan⁻¹x is:

If f(x) = |cos x|, then f'(π/3) = _________.

The function f(x) = [x] (greatest integer function) is discontinuous at all values of x that are __________.

If y = x^x, then dy/dx = _________.

Find the values of 'a' and 'b' such that the function f(x) = { (1 - sin^3 x) / (3 cos^2 x) , if x < π/2 ; a , if x = π/2 ; b(1 - sin x) / ( (π - 2x)^2 ) , if x > π/2 } is continuous at x = π/2.

Show that the function f(x) = |x-1| + |x+1| is continuous but not differentiable at x = 1 and x = -1.