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Maxima and Minima — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMaxima and Minima5 Marks
Question
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ in $[0,3]$
✓

Answer

Given, $f(x) = 3x^4 - 8x^3 + 12x^2 - 48 + 25$
$\Rightarrow f'(x) = 12x^3- 24x^2 + 24x - 28$
For a local maximum or a local minimum, We must have $f'(x) = 0$
$\Rightarrow 12x^3- 24x^2 + 24x - 48 = 0$
$\Rightarrow x^3 - 2x^2+ 2x - 4 = 0$
$\Rightarrow x^2(x - 2) + 2(x - 2) = 0$
$\Rightarrow (x - 2)(x^2 + 2) = 0$
$\Rightarrow x - 2 = 0$ or $(x^2 + 2) = 0$
$\Rightarrow x = 2$
No, real root exists for $ (x^2 + 2) = 0$
Thus, the critical points of $f$ are $0, 2$ and $3.$
Now, $f(0) = 3(0)^4 - 8(0)^3+ 12(0)^2 - 48(0) + 25 = 25$
$f(2) = 3(2)^4 - 8(2)^3 + 12(2)^2 - 48(2) + 25 = -39$
$f(3) = 3(3)^4 - 8(3)^3+ 12(3)^2 - 48(3) + 25 = 1$
Hence, the absolute maximum value when $x = 0$ is $25$ and the absolute minimum value when $x = 2$ is $−39.$

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