Problem 6 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Test the following functions for monotonicity and find stationary points:- i. \(f(x)=2 x^{3}-9 x^{2}+12 x+29\). ii. \(\quad y=x^{3}-3 x^{2}+6 x-17\). iii. \(f(x)=x^{3}-3 x\). iv. \(\quad y=x^{2}(x-3)^{2}\). v. \(\quad f(x)=x^{9}+3 x^{7}+64\) v. \(\quad f(x)=x^{9}+3 x^{7}+64\) viii. \(f(x)=\frac{x}{\ln x}\) ix. \(\quad f(x)=x^{x}\). x. \(\quad f(x)=x^{\frac{1}{x}}\). xi. \(\quad f(x)=x e^{x}\). xii. \(f(x)=x e^{-x}\) xiii. \(f(x)=e^{\frac{1}{x}}\). xiv. \(f(x)=x e^{\frac{1}{x}}\) xv. \(\quad f(x)=\log _{x}(\ln x)\). xvi. \(\quad f(x)=\ln (1+x)-\frac{2 x}{2+x}\). xvii. \(f(x)=x+\sin x\). xviii. \(f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}, \quad 0

Step-by-Step Solution

Answer

In summary, we analyzed three functions \(f(x)=2 x^{3}-9 x^{2}+12 x+29\), \(y=x^{3}-3 x^{2}+6 x-17\), and \(f(x)=x^{3}-3 x\) for monotonicity and stationary points. We found that: Function 1: - Stationary points: \(x=1\) and \(x=2\) - Monotonicity: The function is not monotonic (decreasing, increasing, decreasing) Function 2: - Stationary point: \(x=1\) - Monotonicity: The function is not monotonic (increasing, decreasing) Function 3: - Stationary points: \(x=\sqrt{1}\) and \(x=-\sqrt{1}\) - Monotonicity: The function is not monotonic (increasing, decreasing, increasing)

Step 1: Differentiation

Differentiate the function to obtain the derivative which is \(f'(x)=6x^{2}-18x+12\).

Step 2: Finding the stationary points

Equate the derivative to zero and solve for x. This gives \(6x^{2}-18x+12=0\), whose solutions are \(x=1\) and \(x=2\), the stationary points of the function.

Step 3: Test for Monotonicity

The function is increasing if \(f'(x)>0\) and decreasing if \(f'(x)<0\). Consider the intervals \((-\infty,1),(1,2),(2,\infty)\) then test for the sign of the derivative. The function will be decreasing, increasing and then decreasing respectively if we substitute any value from the mentioned intervals. Therefore, it is not monotonic. Function 2: \(y=x^{3}-3 x^{2}+6 x-17\)

Step 4: Differentiation

Differentiate the function to get \(f'(x)=3x^{2}-6x+6\).

Step 5: Finding the stationary points

Solve \(3x^{2}-6x+6=0\) to get \(x=1\), the stationary point of the function.

Step 6: Test for Monotonicity

Consider the intervals \((-\infty,1)\) and \((1,\infty)\), then test for the sign of the derivative. The function will be increasing and decreasing respectively, indicating that the function is not monotonic. Function 3: \(f(x)=x^{3}-3 x\)

Step 7: Differentiation

The derivative is \(f'(x)=3x^{2}-3\).

Step 8: Finding the stationary points

Solve \(3x^{2}-3=0\) to get \(x=\sqrt{1}\) and \(x=-\sqrt{1}\), the stationary points of the function.

Step 9: Test for Monotonicity

Consider the intervals \((-\infty,-\sqrt{1})\), \((-\sqrt{1},\sqrt{1})\) and \((\sqrt{1},\infty)\), then test for the sign of the derivative. The function will be increasing, decreasing and increasing respectively, indicating the function is not monotonic.

Key Concepts

Stationary Points

Stationary points play a crucial role in understanding the behavior of a function. These are the points where the derivative of a function equals zero. It means that the slope of the tangent to the curve at this point is horizontal. At stationary points, the function can have a local minimum, local maximum, or a point of inflection.

To find stationary points, you take the derivative of the function and solve for when this derivative equals zero. This gives you the x-values of stationary points. For example, for the function \(f(x)=2x^3-9x^2+12x+29\), the derivative is \(f'(x)=6x^2-18x+12\). Setting \(f'(x)=0\) gives the stationary points \(x=1\) and \(x=2\).

• At local maxima, the function changes from increasing to decreasing.
• At local minima, the function changes from decreasing to increasing.
• At points of inflection, the concavity of the function changes but there is no local maxima or minima.

Derivative

The derivative of a function is a fundamental concept in calculus. It measures how the function's value changes as its input changes. You can think of the derivative as the function's "instantaneous rate of change" at a specific point.

Differentiation is the process by which you find this derivative. It gives you a new function (usually denoted by \(f'(x)\) or \(y'\)) that tells you the slope of the original function at any given point. It is vital for identifying stationary points and testing for monotonicity.

• A positive derivative indicates that the function is increasing at that point.
• A negative derivative indicates that the function is decreasing at that point.
• A zero derivative at a point usually means you've found a stationary point.

Knowing how to differentiate basic functions is essential in calculus. For example, the derivative of \(x^3\) is \(3x^2\), and the derivative of \(x^2\) is \(2x\). Complex functions can be differentiated using rules like the product rule, quotient rule, or chain rule.

Test for Monotonicity

Testing for monotonicity involves assessing whether a function is consistently increasing or decreasing over certain intervals. A function is monotonic if it maintains a consistent rise or fall across its entire domain.

To test for monotonicity, we use the derivative of the function. The function is increasing in an interval if \(f'(x)>0\) for all x in that interval. Similarly, it is decreasing if \(f'(x)<0\).

• If a function changes direction, as indicated by signs of \(f'(x)\), it is not monotonic over that interval.
• For example, considering \(f(x)=x^3-3x\), the intervals \((-\infty, -\sqrt{1}), (-\sqrt{1}, \sqrt{1}), (\sqrt{1}, \infty)\) can be tested for monotonicity.

So, checking monotonicity helps us understand the general trends in the function's behavior, finding areas where it grows consistently or decreases consistently.

Polynomial Functions

Polynomial functions are expressions constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions are usually represented as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where the \(a\)'s are constants and \(x\) is a variable.

Polynomials are significant in calculus due to their smoothness and differentiability. For example, the derivative of a polynomial function is another polynomial, which makes them a neat study of derivatives and calculus behavior.

• They can have stationary points where the first derivative is zero.
• The degree of the polynomial influences its number of possible real roots.
• They can exhibit various behaviors based on their coefficients and degree, which impact their graphs.

Understanding polynomial behavior is crucial for identifying stationary points and testing for monotonicity, as seen with functions like \(f(x)=2x^3-9x^2+12x+29\). Their simple, predictable behavior makes them an excellent place to start in calculus.