Consider the function f(x) = { P(x)/sin(x-2), x ≠ 2; 7, x=2 } where P(x) is a polynomial such that P''(x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to…….
Consider the function f(x) = { P(x)/sin(x-2), x ≠ 2; 7, x=2 } where P(x) is a polynomial such that P''(x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to…….
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1 Answer
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P (x) = a (x-2)² + b (x-2) + c.
lim (x→2) P (x)/sin (x-2) = lim (x→2) P (x)/ (x-2) = P' (2) = 7.
P' (x) = 2a (x-2) + b. P' (2) = b = 7.
P' (x) = 2a.
P (3) = a (1)² + b (1) + c = a+b+c = 9.
Continuity at x=2 means lim f (x) = f (2).
lim (x→2) (a (x-2)²+b (x-2)+c)/ (x-2) = P' (2) = b=7. This is given.
The problem states f (2)=7.
P (x) = (x-2) (ax+b) form used in solution. Let's use this.
lim (x→2) (x-2) (ax+b)/sin (x-2) = lim (x→2) ax+b = 2a+b = 7.
P (3) = (3-2) (3a+b) = 3a+b=9.
Solving: a=2, b=3.
P (x) = (x-2) (2x+3).
P (5) = (5-2) (2*5+3) = 3 * 13 = 39.
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