The curve y=a x^3+b x^2+c x+5 touches x -axis at A(-2,0)....

The curve $y=a x^{3}+b x^{2}+c x+5$ touches $x$ -axis at $A(-2,0)$. The curve intersects the $y$ -axis at a point $B$ where its slope equals 3 . The value of $^{\prime} \mathrm{a}$ ' is
(a) $-2$
(b) 2
(c) $\frac{-1}{2}$
(d) $\frac{1}{2}$

Key Concepts

Polynomial Parameter Determination

Given specific conditions such as known points on the curve and specified slopes at those points, one can develop equations by substituting these conditions into the polynomial and its derivative. Solving these simultaneous equations allows for the determination of unknown coefficients within the polynomial.

Repeated Roots and Tangency

When a curve touches the x-axis at a point, the x-value at that point is a repeated (double) root of the polynomial. This means that not only does the function equal zero at that x-value, but its derivative is also zero, indicating that the curve has a horizontal tangent and does not cross the axis at that point.

Differentiation of Polynomial Functions

Differentiation is used to find the slope of a function at a given point. For polynomial functions, the derivative is obtained by applying the power rule to each term. This concept is fundamental when determining conditions for tangency, since a double root implies that both the function and its derivative are zero at that point.

Intersection with the Axes

Understanding where a curve intersects the axes is essential in analyzing its behavior. The x-intercepts are found by setting the function equal to zero, which yields the roots of the equation, while the y-intercept is obtained by evaluating the function at x = 0. These intersections provide crucial conditions for solving problems involving the determination of the function's parameters.

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