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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives4 Marks
Question
Find the points at which the function f given by f(x) = (x - 2)4 (x + 1)3 has
  1. local maxima
  2. local minima
  3. point of inflexion
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Answer

It is given that function is f(x) = (x - 2)4 (x + 1)3 
$\Rightarrow$ f'(x) = 4(x - 2)3 (x + 1)3 + 3(x + 1)2 (x - 2)4 
= (x - 2)3(x + 1)2 [4(x + 1) + 3(x - 2)]
= (x - 2)3(x + 1)2 (7x - 2)
Now, f'(x) = 0
$\Rightarrow$ x = -1 and x = $\frac{2}{7}$ or x = 2
Now, for values of x close to $\frac{2}{7}$ and to the left of $\frac{2}{7}$
f'(x) > 0
Also, for values of x close to $\frac{2}{7}$ and to the right of $\frac{2}{7}$, f'(x) < 0.
Then, x = $\frac{2}{7}$ is the point of local maxima.
Now, for values of x close to 2 and to the left of 2, f'(x) < 0.
Also, for values of x close to 2 and to the right of 2, f'(x) > 0.
Then, x = 2 is the point of local minima.
Now, as the value of x varies through -1, f'(x) does not changes its sign.
Then, x = -1 is the point of inflexion.

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