Let f(x) be continuous and differentiable function for all reals.… - Vidyadip

MCQ

Let $f(x)$ be continuous and differentiable function for all reals.

$f(x + y)\, = \,f(x) - 3xy + f(y).$ If $\mathop {\lim }\limits_{h \to 0} \frac{{f(h)}}{h} = 7$ then value of $f'(x)$ is-

A
$-3x$
B
$7$
✓
$-3x+7$
D
$2f(x)+7$

✓

Answer

Correct option: C.

$-3x+7$

c
${{\rm{f}}^\prime }({\rm{x}}) = \mathop {\lim }\limits_{h \to 0} \frac{{{\rm{f}}({\rm{x}} + {\rm{h}}) - {\rm{f}}({\rm{x}})}}{{\rm{h}}}$

$ = \mathop {\lim }\limits_{h \to 0} \frac{{f(x) - 3xh + f(h) - f(x)}}{h}$

$ = \mathop {\lim }\limits_{h \to 0} \left( { - 3x + 7 + \frac{{f(h)}}{h}} \right)$

$=-3 x+7$

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