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Model Paper 2 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 21 Mark
Question
Assertion $(A):$ The absolute maximum value of the function $2 x^3-24 x$ in the interval$ [1, 3]$ is $89.$
Reason $(R):$ The absolute maximum value of the function can be obtained from the value of the function at critical points and at boundary points.
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Answer

Let $f(x)=2 x^3-24 x$
$ \Rightarrow f^{\prime}(x)=6 x^2-24=6\left(x^2-4\right)$
$=6(x+2)(x-2)$
For maxima or minima put $f^{\prime}(x)=0$.
$\Rightarrow 6(x+2)(x-2)=0$
$ \Rightarrow x=2,-2 $
We first consider the interval $[1,3]$.
So, we have to evaluate the value of $f$
at the critical point $x=2 \in[1,3]$ and at the end points of $[1,3]$.
At $x=1, f(1)=2 \times 1^3-24 \times 1=-22$
At $x=2, f(2)=2 \times 2^3-24 \times 2=-32$
At $x=3, f(3)=2 \times 3^3-24 \times 3=-18$
$\therefore$ The absolute maximum value of $f ( x )$ in the interval $[1,3]$ is -18 occurring at $x =3$.
Hence, Assertion is false and Reason is true.

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