Let the function f- - be defined by fx-x3-x2-x-1sin x and let g- - be an arbitrary function- Let fg- - - be the product function defined by fgx-fxgx-Then which of the following statements is-are TRUE-

F:ℝ→ ℝ (A):f(x)=x^3 x^2+(x 1)sin x; g:ℝ→ ℝ h(x)=f(x)· g(x)=[x^3 x^2+(x 1)sin x]· g(x) h'(1^+)=lim_h → 0[(1+h)^3 (1+h)^2+h·sin(1+h)] g(1+h)h =lim_h → 0(1+h^3+3h+ ...

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Byju's Answer

Standard XII

Mathematics

Theorems for Continuity

Let the funct...

Question

Let the function $f : R \to R$ be defined by $f (x) = x^{3} - x^{2} + (x - 1) sin x$ and let $g : R \to R$ be an arbitrary function. Let $f g : R \to R$ be the product function defined by $(f g) (x) = f (x) g (x) .$ Then which of the following statements is/are TRUE?

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Solution

$f : R \to R$
$(A) : f (x) = x^{3} - x^{2} + (x - 1) sin x; g : R \to R$
$h (x) = f (x) \cdot g (x) = [x^{3} - x^{2} + (x - 1) sin x] \cdot g (x)$
$h^{'} (1^{+}) = lim h \to 0 \frac{[(1 + h)^{3} - (1 + h)^{2} + h \cdot sin (1 + h)] g (1 + h)}{h}$
$= lim h \to 0 \frac{(1 + h^{3} + 3 h + 3 h^{2} - 1 - h^{2} - 2 h + h sin (1 + h)) g (1 + h)}{h}$
$= lim h \to 0 \frac{(h^{3} + 2 h^{2} + h + h sin (1 + h)) g (1 + h)}{h}$
$= lim h \to 0 (1 + sin (1 + h)) g (1 + h)$
$h^{'} (1^{-}) = lim h \to 0 \frac{[(1 - h)^{3} - (1 - h)^{2} + (- h) \cdot sin (1 - h)] g (1 - h)}{- h}$
$= lim h \to 0 \frac{(1 - h^{3} - 3 h + 3 h^{2} - 1 - h^{2} + 2 h - h sin (1 - h)) g (1 - h)}{- h}$
$= lim h \to 0 (1 + sin (1 - h)) g (1 - h)$

as $g (x)$ is continuous at $x = 1$
$∴ g (1 + h) = g (1 - h) = g (1)$
$h^{'} (1^{+}) = h^{'} (1^{-}) = (1 + sin 1) g (1)$
$^{'} A^{'}$ is correct

$C$ is always true as
$h (x) = f (x) \cdot g (x) \Rightarrow h^{'} (x) = f^{'} (x) \cdot g (x) + f (x) \cdot g^{'} (x)$

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Let the function f:R→R be defined by f(x)=x3−x2+(x−1)sinx and let g:R→R be an arbitrary function. Let fg:R→R be the product function defined by (fg)(x)=f(x)g(x).Then which of the following statements is/are TRUE?

Let the function $f : R \to R$ be defined by $f (x) = x^{3} - x^{2} + (x - 1) sin x$ and let $g : R \to R$ be an arbitrary function. Let $f g : R \to R$ be the product function defined by $(f g) (x) = f (x) g (x) .$ Then which of the following statements is/are TRUE?