Let's break down the exercise step by step.

Given:

The function $f(x) = x(1 - x)$.

Part 1:

Show that for all $x \in \mathbb{R}$, $f(x) \leq \frac{1}{4}$.

• First, express the function clearly: $f(x) = x(1 - x) = x - x^2$

  To find the maximum value of $f(x)$, we can take the derivative and find its critical points.

  $f'(x) = 1 - 2x$

  Set $f'(x) = 0$ to find the critical point: $1 - 2x = 0 \implies x = \frac{1}{2}$

  Now evaluate $f(x)$ at $x = \frac{1}{2}$: $f\left( \frac{1}{2} \right) = \frac{1}{2} \left( 1 - \frac{1}{2} \right) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

  Since the parabola $f(x) = x - x^2$ opens downward (as the coefficient of $x^2$ is negative), this is the maximum value. Therefore, for all $x \in \mathbb{R}$, $f(x) \leq \frac{1}{4}$.

Part 2:

Deduce that the function $f$ attains a maximum at $x = \frac{1}{2}$.

From Part 1, we have already found that the maximum value of $f(x)$ is $\frac{1}{4}$, and this occurs at $x = \frac{1}{2}$. Therefore, $f(x)$ has a maximum at $x = \frac{1}{2}$.

Part 3:

Demonstrate that $f(x) = \frac{1}{4}(x - \frac{1}{2})^2$ and deduce that the function is increasing on $\left( -\infty, \frac{1}{2} \right]$ and decreasing on $\left[ \frac{1}{2}, \infty \right)$.

• Let's rewrite $f(x) = x - x^2$ in the form $f(x) = \frac{1}{4} - \frac{1}{4}(x - \frac{1}{2})^2$.

  $f(x) = x - x^2 = \frac{1}{4} - \frac{1}{4}(x - \frac{1}{2})^2$

  This is a standard form of a downward-opening parabola. The vertex form of the parabola shows that $f(x)$ reaches its maximum when $x = \frac{1}{2}$, and the value of the function decreases as we move away from $x = \frac{1}{2}$.

• Since the parabola opens downward, the function is:

  • Increasing on the interval $\left( -\infty, \frac{1}{2} \right]$,
  • Decreasing on the interval $\left[ \frac{1}{2}, \infty \right)$.

Summary:

• The maximum value of $f(x)$ is $\frac{1}{4}$, and it occurs at $x = \frac{1}{2}$.
• The function is increasing on $\left( -\infty, \frac{1}{2} \right]$ and decreasing on $\left[ \frac{1}{2}, \infty \right)$.

Do you want more details on any step, or explanations of certain concepts?

Related Questions:

1. How do you find the maximum or minimum of a quadratic function?
2. What does the derivative tell us about the increasing or decreasing nature of a function?
3. Can the maximum value of a quadratic function occur at a point other than the vertex?
4. How can you express a quadratic function in vertex form?
5. How does completing the square help in finding the maximum or minimum of a function?

Tip: For quadratic functions, the sign of the $x^2$-coefficient determines whether the parabola opens upwards or downwards (positive opens upwards, negative opens downwards).