If the slope of the tangent to the curve y=ax | vedclass

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(A) The curve $y=ax^3+bx+4$ passes through the point $(2, 14)$. Substituting these coordinates into the equation: $14 = a(2)^3 + b(2) + 4$ $14 = 8a +

Vedclass · Accepted Answer

$2, -3$

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(A) The curve $y=ax^3+bx+4$ passes through the point $(2, 14)$. Substituting these coordinates into the equation: $14 = a(2)^3 + b(2) + 4$ $14 = 8a + 2b + 4$ $10 = 8a + 2b$ $5 = 4a + b$  --- $(i)$ The slope of the tangent to the curve is given by the derivative $\frac{dy}{dx}$: $\frac{dy}{dx} = 3ax^2 + b$ At the point $(2, 14)$, the slope is $21$: $21 = 3a(2)^2 + b$ $21 = 12a + b$ --- $(ii)$ Subtracting equation $(i)$ from equation $(ii)$: $(12a + b) - (4a + b) = 21 - 5$ $8a = 16$ $a = 2$ Substituting $a = 2$ into equation $(i)$: $5 = 4(2) + b$ $5 = 8 + b$ $b = -3$ Thus, the values are $a = 2$ and $b = -3$.

If the slope of the tangent to the curve $y=ax^3+bx+4$ at the point $(2, 14)$ is $21$, then the values of $a$ and $b$ are respectively:

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