SATHEE: Application of Derivatives 4 Question 19

Application of Derivatives 4 Question 19

20. The total number of local maxima and local minima of the function $f (x) = \begin{array}{cc} (2 + x)^{3}, & - 3 < x \leq - 1 x^{\frac{2}{3}}, & - 1 < x < 2 \end{array}$ is

(2008, 3M)

(a) 0

(b) 1

(d) 3

Show Answer

Answer:

Correct Answer: 20. (d)

Solution:

Given, $f (x) = \begin{array}{cc} (2 + x)^{3}, & if - 3 < x \leq - 1 x^{2 / 3} & if - 1 < x < 2 \end{array}$

$\Rightarrow f^{'} (x) = \begin{array}{ll} 3 (x + 2)^{2}, & if - 3 < x \leq - 1 \frac{2}{3} x^{- \frac{1}{3}} & , if - 1 < x < 2 \end{array}$

Clearly, $f^{'} (x)$ changes its sign at $x = - 1$ from $+ v e$ to $- v e$ and so $f (x)$ has local maxima at $x = - 1$ .

Also, $f^{'} (0)$ does not exist but $f^{'} (0) < 0$ and $f^{'} (0^{+}) < 0$ . It can only be inferred that $f (x)$ has a possibility of a $minima$ at $x = 0$ . Hence, the given function has one local maxima at $x = - 1$ and one local minima at $x = 0$ .

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Application of Derivatives 4 Question 19

20. The total number of local maxima and local minima of the function f(x)=(2+x)3,−3<x≤−1 x23,−1<x<2 is

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20. The total number of local maxima and local minima of the function $f (x) = \begin{array}{cc} (2 + x)^{3}, & - 3 < x \leq - 1 x^{\frac{2}{3}}, & - 1 < x < 2 \end{array}$ is