The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+5$, is:
Answer & explanation
Correct answer: option 3
The function $f(x) = x^3$ increases for all x and the function $g(x) = 6x^2 +15x + 5$ increases if
$g'(x)>0⇒12x+15>0⇒x>-\frac{5}{4}$
Thus, f(x) and g(x) both increase for $x>-\frac{5}{4}$
It is given that f(x) increases less rapidly than g(x), therefore the function $\phi(x) = f(x) - g(x)$ is decreasing function, which implies that $\phi{'}(x) <0$.
$⇒ 3x^2 -12x -15 < 0 ⇒ x^2 - 4x - 5 < 0 ⇒ (x - 5)(x +1) < 0$
$⇒-1 < x < 5$
Hence, $x^3$ increases less rapidly than $6x^2 +15x + 5$ on the interval (-1, 5).