If f(x) = x2 – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is
Explanation:
f(x) = x2 – 7x and g(x) = x + 3
⇒ f(g(x)) – 3x = (g(x))2 – 7g(x) – 3x
⇒ f(g(x)) - 3x = (x + 3)2 – 7(x + 3) - 3x
⇒ f(g(x)) - 3x = x2 + 6x + 9 – 7x – 21 – 3x
⇒ f(g(x)) = x2 - 4x – 12
f(g(x)) is a quadratic equation. Least value of ax2 + bx + c occurs at x = -b/2a
∴ Minimum value of x2 - 4x – 12 occurs at x = -(-4)/2 = 2
∴ Minimum value of f(g(x)) = 4 - 8 - 12 = -16.
Hence, option (b).
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