Investigate - x3 - x2 - 10x - x - 0- - 3sin x -x - 0 - at x - 0 for local maxima - minima-

{[ x^3 + x^2 + 10x , amp; x lt; 0; 3sin x amp; ,x ≥ 0 ]. for x ≥ 0 f( x ) = #160; 3sin x f'( x ) = #160; 3cos ...

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Standard XII

Physics

Double Differentiation

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Question

Investigate ${\begin{matrix} x^{3} + x^{2} + 10 x, & x < 0 - 3 sin x & , x \geq 0 \end{matrix}$ at $x = 0$ for local maxima / minima.

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Solution

${\begin{matrix} x^{3} + x^{2} + 10 x, & x < 0 - 3 sin x & , x \geq 0 \end{matrix}$
$f o r x \geq 0$
$f (x) = - 3 sin x$
$f^{'} (x) = - 3 cos x$
$f^{'} (0) = - 3$
$f o r x < 0$
$f (x) = x^{2} + x^{2} + 10 x$
$f^{'} (x) = 3 x^{2} + 2 x + 10$
$f^{'} (0) = 10$
$f^{'} (x) > f o r x < 0 a n d f^{'} (x) < 0 f o r x \geq 0$
$f (x)$ has maxima at $x = 0$

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Find the local maxima and local minima, if any of the following functions. Also, find the local maximum and the local minimum values, as the case may be as follows.
$f (x) = x^{2}$

$g (x) = x^{3} - 3 x$

$h (x) = s i n x + c o s x, 0 < x < \frac{π}{2}$

$f (x) = s i n x - c o s x, 0 < x < 2 π$

$f (x) = x^{3} - 6 x^{2} + 9 x + 15$

$g (x) = \frac{x}{2} + \frac{2}{x}, x > 0$

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Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i). f(x) = x² (ii). g(x) = x³ − 3x

(iii). h(x) = sinx + cos, 0 < (iv). f(x) = sinx − cos x, 0 < x < 2π

(v). f(x) = x³ − 6x²+ 9x + 15

(vi).

(vii).

(viii).

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Investigate {x3+x2+10x,x<0−3sinx,x≥0 at x=0 for local maxima / minima.

{x3+x2+10x,x<0−3sinx,x≥0forx≥0f(x)=−3sinxf′(x)=−3cosxf′(0)=−3forx<0f(x)=x2+x2+10xf′(x)=3x2+2x+10f′(0)=10f′(x)>forx<0andf′(x)<0forx≥0f(x) has maxima at x=0

Investigate ${\begin{matrix} x^{3} + x^{2} + 10 x, & x < 0 - 3 sin x & , x \geq 0 \end{matrix}$ at $x = 0$ for local maxima / minima.