Consider \( f \), \( g \) and \( h \) be three real valued differentiable functions defined on \( R \).
Let \( g(x)=x^{3}+g^{\prime \prime}(1) x^{2}+\left(3 g^{\prime}(1)-g^{\prime \prime}(1)-1\right) x+3 g^{\prime}(1) \) \( f(x)=x g(x)-12 x+1 \)
and \( f(x)=(h(x))^{2} \), where \( h(0)=1 \)
Which of the following is/are true for the function \( y=g(x) ? \)
(a) \( g(x) \) monotonically decreases inEREDEDDD
\( \left(-\infty, 2-\frac{1}{\sqrt{3}}\right) \cup\left(2+\frac{1}{\sqrt{3}}, \infty\right) \)
(b) \( g(x) \) monotonically increases in \( \left[2-\frac{1}{\sqrt{3}}, 2+\frac{1}{\sqrt{3}}\right) \)
(c) There exists exactly one tangent to \( y=g(x) \) which is parallel to the chord joining the points \( (1, g(1)) \) and \( (3, g(3)) \)
(d) There exists exactly two distinct lagrange's mean value in \( (0,4) \) for the function \( y=g(x) \)
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