Problem 15 Find the turning points on \(y=f... [FREE SOLUTION] (2024)

Short Answer

second derivative test

The second derivative test helps us determine the nature of turning points (local maxima, minima, or points of inflection) on a function using the second derivative, denoted as \( \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} \).
When \( \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} > 0 \) at a critical point, the function has a local minimum there.
Alternatively, if \( \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} < 0 \), the function exhibits a local maximum.
A zero value might mean a point of inflection, but further analysis is needed.
In our problem:
Given \( \frac{\text{d}^2 y}{\text{d} x^2} = 3x - 2 \)
We need to check the values at the turning points \( x = 0 \) and \( x = 1 \).
- For \( x = 0 \): \( \frac{\text{d}^2 y}{\text{d} x^2} = 3(0) - 2 = -2 \), indicating a local maximum.
- For \( x = 1 \): \( \frac{\text{d}^2 y}{\text{d} x^2} = 3(1) - 2 = 1 \), indicating a local minimum.

integration

Integration is the process of finding the antiderivative or the area under the curve.
To find the first derivative \( \frac{\text{d} y}{\text{d} x} \) from \( \frac{\text{d}^2 y}{\text{d} x^2} \), we integrate:
Given \( \frac{\text{d}^2 y}{\text{d} x^2} = 3x - 2 \), we integrate:
\[ \frac{\text{d} y}{\text{d} x} = \int (3x - 2)\,dx = \frac{3}{2}x^2 - 2x + C \] where \( C \) is a constant that we determine using initial conditions or additional information.

quadratic functions

Quadratic functions are polynomial functions of degree 2, typically in the form \( ax^2 + bx + c \).
In this exercise, \( \frac{\text{d} y}{\text{d} x} \) is a quadratic function.
We derived \( \frac{\text{d} y}{\text{d} x} \) as \( \frac{3}{2}x^2 - 2x + C \).
Since we know the roots (zero points) of this function are \( x = 0 \) and \( x = 1 \), we express it as:
\[ \frac{\text{d} y}{\text{d} x} = kx(x-1) \] To match this to our derived equation and find \( k \), we solve:
\( kx^2 - kx = \frac{3}{2}x^2 - 2x + C \).

roots of equations

Roots of equations are the values of \( x \) that make an equation equal to zero.
For \( \frac{\text{d} y}{\text{d} x} = 0 \), solving this quadratic equation gives the roots of the function.
Given \( \frac{\text{d} y}{\text{d} x} = \frac{3}{2}x^2 - 2x + C \) with roots \( x = 0 \) and \( x = 1 \), we identify crucial points to further analyze.
These roots indicate potential turning points when the first derivative is zero.
We analyze these roots using the second derivative test to determine if they are maxima, minima, or points of inflection.