Question

The function y = f(x) is defined by x = 2t – |t|,

y = t² + t |t|, t Î R in the interval x Î [– 1, 1] then

• A. f(x) is continuous every where
• B. f(x) is not continuous at x = 0
• C. f(x) is continuous but not derivable at x = 0
• D. f(x) is constant function

Answer 1

Answer: (A)

f(x) is continuous every where

Answer 2

When t ³ 0 x = 2t – t = t

y = t² + t² = 2t²

Now relation between x & y

y = 2x² , t ³ 0, x ³ 0

y = 2x², x ³ 0

when t < 0

x = 2t – (–t) = 3t

y ̃ t² + t (–t) = 0

Now relation between x & y

y = 0, t < 0, x < 0

y = 0, x < 0

so that we can write

f(x) = $\begin{Bmatrix} 2x^{2} & 0 \leq x \leq 1 \\ 0 & –1 \leq x < 0 \end{Bmatrix}$

clearly f(x) is continuous and diff. everywhere.