let f be a function defined by: f(x) = x^2, for x <= 1 f(x) = ax^2 + bx + c,…
1996
let f be a function defined by:
f(x) = x^2, for x <= 1
f(x) = ax^2 + bx + c, for 1 < x <= 2
f(x) = x + d, for x > 2
find the values of a, b, c and d so that f is continuous and differentiable everywhere on the real line.
- A.
a = -0.5, b = 3, c = -1.5, d = 0.5
- B.
a = -1, b = 1, c = 1, d = 1
- C.
a = -2, b = 1, c = -1.5, d = 0.5
- D.
None of These
Show answer & explanation
Correct answer: A
for continuity at x = 1, the middle piece must match x^2, so a + b + c = 1.
for differentiability at x = 1, derivatives must match, so 2a + b = 2.
for differentiability at x = 2, the derivative of the middle piece must match the derivative of x + d, so 4a + b = 1.
solving 2a + b = 2 and 4a + b = 1 gives a = -0.5 and b = 3. then a + b + c = 1 gives c = -1.5.
for continuity at x = 2, 4a + 2b + c = 2 + d. substituting the values gives 2.5 = 2 + d, so d = 0.5.
therefore, a = -0.5, b = 3, c = -1.5 and d = 0.5.