tmua.me Extreme Set B Paper 2 · Question 18

For a real constant $k$ , define

f_k(x)=x^3-kx^2+10x.

Consider the statements

S: $k\geq13/2$ .
T: For every pair of distinct positive real numbers $a,b$ satisfying

$\displaystyle ab=1\qquad\text{and}\qquad f_k(a)=f_k(b),$

the tangents to $y=f_k(x)$ at $x=a$ and $x=b$ both have strictly positive gradient.

Is S necessary and/or sufficient for T?

AS is neither necessary nor sufficient for T

BS is necessary but not sufficient for T

CS is sufficient but not necessary for T

DS is necessary and sufficient for T

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