<s>[INST] interval in which $h(x)$ is increasing If $h(x)=2g(2x^3-3x^2)+g(6x^2-4x^3-3)\forall x\in\mathbb{R}$ and $g''(x)>0\forall x\in\mathbb{R}$. Then find values of $x$ for which $h(x)$ is increasing
solution i try
$h(x)=2g'(2x^3-3x^2)(6x^2-6x)-g'(6x^2-4x^3-3)(12x^2-12x)$
From $g''(x)>0$ implies $g'(x)$ is increasing function.
Not be able to solve ahead  Help required
 [/INST] Jacky. 

$$ h(x) = 2g(2x^3−3x^2)+g(6x^2−4x^3−3) $$
Now let $g(2x^3−3x^2) = g(a)$ and $g(6x^2−4x^3−3)=g(b).$  So the 1st deriv of $h(x)$ is :
$$h'(x) = 12 \left[ g'(a)(x^2 - x) + g'(b)(x-x^2) \right] $$
$$ = 12 (x^2-x)\left[ g'(a) - g'(b) \right] $$
Now $h'(x)$ will be positive when 
$$ x^2-x > 0 \:\: \text{and} \:\: g'(a)>g'(b) $$
$$ \cup $$
$$ x^2-x < 0 \:\: \text{and} \:\: g'(a) < g'(b) $$
Since we know that $g'(x)$ is increasing, then $g'(a) > g'(b)$ is achieved when $a > b$, or $2x^3-3x^2 > 6x^2-4x^3-3$. 
so now $h'(x)$ will be positive when 
$$ x^2-x > 0 \:\: \text{and} \:\: 2x^3-3x^3 > 6x^2-4x^3-3 $$
$$ \cup $$
$$ x^2-x < 0 \:\: \text{and} \:\: 2x^3-3x^3 < 6x^2-4x^3-3 $$
Can you continue from here?
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