f-x-ax-2-bx-1-x-0-cx-2-d-0-lt-x-1-2-bx-d-1-2-lt-x-1-f-x-is-continuous-on-1-1-prove-d-0-c-2b-

Question Number 87799 by M±th+et£s last updated on 06/Apr/20

f (x) = {\begin{cases} {a x}^{2} + b x - 1 ⩽ x ⩽ 0 \\ {c x}^{2} + d 0 < x ⩽ \frac{1}{2} \\ b x + d \frac{1}{2} < x ⩽ 1 \end{cases}

f (x) i s c o n t i n u o u s o n [- 1, 1]

p r o v e d = 0

c = 2 b

Commented by john santu last updated on 06/Apr/20

\lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{+}} f (x)

\lim_{x \to 0^{-}} ({ax}^{2} + bx) \lim_{x \to 0^{+}} ({cx}^{2} + d)

0 = d

\lim_{x \to 0 . 5^{-}} ({cx}^{2} + d) = \lim_{x \to 0 . 5^{+}} (bx + d)

\frac{1}{4} c = \frac{1}{2} b \Rightarrow c = 2 b

Commented by M±th+et£s last updated on 06/Apr/20

g o d b l e s s y o u s i r