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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
The interval in which the function ${x^3} $ increases less rapidly than $6{x^2} + 15x + 5$, is
  • A
    $( - \infty ,\, - 1)$
  • B
    $(-5 , 1)$
  • ✓
    $(-1 ,5)$
  • D
    $(5 , \infty )$
✓

Answer

Correct option: C.
$(-1 ,5)$
c
(c) The function $f(x) = {x^3}$ increases for all $x$ and the function $g(x) = 6{x^2} + 15x + 5$ increases, if $g'(x) > 0 $

$\Rightarrow 12x + 15 > 0 \Rightarrow x > - \frac{5}{4}$.

Thus $f(x)$ and $g(x)$ both increases 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 $\phi '(x) < 0$

==> $3{x^2} - 12x - 15 < 0 \Rightarrow - 1 < x < 5$.

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