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Application of Derivatives — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSApplication of Derivatives1 Mark
Question
Prove that the function given by $f\left( x \right) = {x^3} - 3{x^2} + 3x - 100$ is increasing in R.
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Answer

Given: $f\left( x \right) = {x^3} - 3{x^2} + 3x - 100$

$\Rightarrow f'\left( x \right) = 3{x^2} - 6x + 3 = 3\left( {{x^2} - 2x + 1} \right)$

$\Rightarrow f'\left( x \right) = 3{\left( {x - 1} \right)^2} \geqslant 0$ for all x in R.

Therefore, $f\left( x \right)$ is increasing in R.

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