If f(x+y)=f(x)+f(y)+6xy(x+y)∀x,y∈ R and f'(0)=−8 then which of the following Statements is TRUE?

Use, f(0)=0,f'(0)=−8,f'(x)=limh→0f(x+h)−f(x)h=limh→0(f(h)h+6x(x+h))=f'(0)+6x2 ∴f'(x)=6x2−8

Use, f(0)=0,f'(0)=−8,f'(x)=limh→0f(x+h)−f(x)h=limh→0(f(h)h+6x(x+h))=f'(0)+6x2 ∴f'(x)=6x2−8

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If $f (x + y) = f (x) + f (y) + 6 x y (x + y) \forall x, y \in R a n d f' (0) = - 8$ then which of the following Statements is TRUE?

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The slopes of the tangents at the points where $y = f (x)$ cuts X-axis have same sign

The tangents at the points where $y = f (x)$ cuts X-axis are parallel

The slope of the tangent where x=1 is equal to -2

The slope of the tangent is zero when $x = \pm 4 / 3$

answer is C.

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Use, $f (0) = 0, f' (0) = - 8, f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$= \lim_{h \to 0} (\frac{f (h)}{h} + 6 x (x + h)) = f' (0) + 6 x^{2} ∴ f' (x) = 6 x^{2} - 8$

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