Q.The equation of the path traced by a roller-coaster is given by the polynomial $f(x) = a(x + 9)(x + 1)(x - 3)$. If the roller-coaster crosses y-axis at a point $(0, -1)$, answer the following :

Concept: Polynomial Roots and Derivatives — the polynomial is given in factored form; the y-intercept fixes the leading coefficient, and the second derivative measures concavity.

Step 1 – Find $a$ using the y-intercept.

At $x = 0$, $f(0) = a(9)(1)(-3) = -27a$.

Given $f(0) = -1$, so $-27a = -1 \implies a = \frac{1}{27}$.

Step 2 – Write $f(x)$ and differentiate.

$f(x) = \frac{1}{27}(x+9)(x+1)(x-3)$.

First expand: $(x+9)(x+1)(x-3) = (x^2+10x+9)(x-3) = x^3 + 7x^2 - 21x - 27$.

Thus $f(x) = \frac{1}{27}(x^3 + 7x^2 - 21x - 27)$. …

(i) Find $a$. The curve crosses the $y$-axis at $(0,-1)$, so $f(0)=-1$:

Setting $-27a=-1$ gives

(ii) Find $f''(1)$. Expand $f(x)=\dfrac{1}{27}(x+9)(x+1)(x-3)$:

Differentiate: …

Method: Finding an Unknown Coefficient from a Data Point, Then Evaluating a Higher-Order Derivative

Use this method for problems that give a polynomial in factored form with one unknown constant, supply one known point on the curve, and then ask for a derivative value.

Steps

Step 1: Use the given point to solve for the unknown constant

Substitute the known $(x,y)$ pair directly into the factored polynomial and solve the resulting linear equation for the unknown coefficient — this is almost always the fastest way, since the other factors become plain numbers once $x$ is fixed.

Step 2: Expand the polynomial into standard form

Multiply out the factors step by step (pairing two factors first, then multiplying by the third) to get a single polynomial in $x$ with the now-known coefficient included. …

Common Mistakes

Mistake 1: Sign error when substituting $x=0$ into the negative factor

In $f(0) = a(0+9)(0+1)(0-3)$, the last factor is $(0-3) = -3$, a negative number — a student rushing through arithmetic can easily drop or misplace this negative sign, turning $-27a$ into $27a$ and getting the wrong value of $a$ (with the wrong sign). Correct approach: evaluate each factor separately and explicitly before multiplying, especially when one of them is negative.

Mistake 2: Algebra slip while expanding the triple product

Multiplying three binomials in one's head, rather than pairing two first and then multiplying by the third, is a common source of a dropped or mis-signed term (e.g. missing the middle term when combining $(x^2+10x+9)$ with $(x-3)$). Correct approach: expand two factors at a time, write out the intermediate quadratic explicitly, and only then multiply by the remaining factor. …