Question

The value of 'a' for which all extremums of function \(f(x)=x^{3}-3 a x^{2}+3\left(a^{2}-1\right) x+1\), lie in the interval \((-2,4)\) is (1) \((3,4)\) (2) \((-1,3)\) (3) \((-3,-1)\) (4) none of these

Short answer

The interval is (-1,3).

Step-by-step solution

- Find the derivative of the function

To locate the extremums of the function, first find the derivative of the function \[ f(x) = x^3 - 3ax^2 + 3(a^2 - 1)x + 1 \]The derivative, f'(x), is given by:\[ f'(x) = 3x^2 - 6ax + 3(a^2 - 1) \]

- Solve for critical points

Set the first derivative equal to zero to find the critical points:\[ 3x^2 - 6ax + 3(a^2 - 1) = 0 \]Divide everything by 3 for simplicity:\[ x^2 - 2ax + (a^2 - 1) = 0 \]This is a quadratic equation in terms of x. Use the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where a=1, b=-2a, and c=a^2-1. Substituting these values we get:\[ x = \frac{2a \pm \sqrt{4a^2 - 4(a^2-1)}}{2} \]Simplify the expression under the square root:\[ x = \frac{2a \pm \sqrt{4}}{2} \]\[ x = \frac{2a \pm 2}{2} \]\[ x = a \pm 1 \]

- Ensuring critical points are within the interval

The critical points must lie in the interval (-2, 4). Therefore, we have:\[ -2 < a-1 < 4 \] and \[ -2 < a+1 < 4 \]Solve these inequalities separately:For \[ -2 < a-1 < 4 \]:\[ -2 + 1 < a < 4 + 1 \]\[ -1 < a < 5 \]For \[ -2 < a+1 < 4 \]:\[ -2 - 1 < a < 4 - 1 \]\[ -3 < a < 3 \]Combining these two inequalities gives:\[ -1 < a < 3 \]

- Select the right interval

The correct interval that satisfies the condition from the given options is:Option (2) \[(-1, 3)\]

Key concepts

Derivatives

A derivative is a tool in calculus that helps us understand how a function changes. It represents the rate at which the function's value changes as its input changes. For a function's graph, the derivative at a point corresponds to the slope of the tangent line to the graph at that point. To find the derivative of a function, we apply different rules such as the power rule, the product rule, and the chain rule. In our problem, we started by finding the derivative of the given function. This step is crucial because it helps us locate the critical points where the function's slope is zero.

Critical Points

Critical points are where a function's derivative is zero or undefined. These points can indicate where the function reaches local maxima, minima, or saddle points. To find these points, we first derive the function and then solve for when this derivative equals zero. In the problem, we derived the function and set the derivative equal to zero to get a quadratic equation. Solving this equation using the quadratic formula gives us the critical points. These points are a crucial part of analyzing the function because they tell us where potential extremums (maximum or minimum values) occur.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, usually in the form of: \[ax^2 + bx + c = 0\]. To solve these equations, we use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula helps find the roots of the quadratic equation. In our problem, after finding the derivative and setting it to zero, we simplified it to a quadratic equation with the variable x. By applying the quadratic formula, we found the critical points as\[x = a \pm 1\]. This step was essential for determining the values of x where the rate of change of the function is zero.

Interval Analysis

Interval analysis involves checking which ranges of a variable satisfy certain conditions. This is useful in understanding the behavior of functions within specific intervals. In this problem, we analyzed the interval of the critical points. We ensured that our critical points proposed in the interval (-2, 4). This meant solving inequalities to find the range for 'a' such that (a-1) and (a+1) are both within (-2, 4). Combining the solution ranges for these inequalities gave us the final range, ensuring that all extremums of the function lie within the given interval. Interval analysis allowed us to narrow down to the correct value of 'a' among the provided options.