The smallest possible value of a in the given function is 1/5.
Value of f(3)
From the given function, the value of f(3) is calculated as follows;
f(x)=ax² + [tex]\frac{2}{a} x[/tex]
f(3) = a(3)² + [tex]\frac{2}{a}(3)[/tex]
f(3) = 9a + 6/a
Value of f(2)
f(x)=ax² + [tex]\frac{2}{a} x[/tex]
f(2) = a(2²) + [tex]\frac{2}{a} (2)[/tex]
f(2) = 4a + 4/a
f(3) – f(2) = 11
(9a + 6/a) - ( 4a + 4/a) = 11
9a + 6/a - 4a - 4/a = 11
5a + 2/a = 11
multiply through with "a"
5a² + 2 = 11a
5a² - 11a + 2 = 0
5a² - 10a - a + 2 = 0
5a(a - 2) - 1(a - 2) = 0
(5a - 1)(a - 2) = 0
a = 2, or
5a = 1
a = 1/5
Thus, the smallest possible value of a in the given function is 1/5.
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