Consider the polynomials f(x) = ax2 + bx + c, where a > 0, b, c are real, g(x) = -2x. If f(x) cuts the x-axis at (-2, 0) and g(x) passes through (a, b), then the minimum value of f(x) + 9a + 1 is
Explanation:
g(x) = -2x passes through (a, b) ⇒ b = -2a ...(1)
Now, f(x) = ax2 + bx + c ⇒ f(x) = ax2 - 2ax + c [from (1)]
f(x) passes through (2, 0) ⇒ a(-2)2 - 2a(-2) + c = 0 ⇒ 8a + c = 0 ⇒ c = -8a
∴ f(x) = ax2 - 2ax - 8a
Now, we have to find the least value of f(x) + 9a + 1 = ax2 - 2ax - 8a + 9a + 1 = ax2 - 2ax + a + 1 = a(x2 - 2x + 1) + 1 = a(x - 1)2 + 1
Least value of this expression will be 1, when x = 1.
Hence, option (b).
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