The given problem involves the function $f(x) = \frac{1}{2x^2 - 3x + 6}$ and requires the following:

(a)

Using MATLAB to graph $f(x)$ on the interval $[-3, 3]$, and zooming in to find the point where $f(x)$ is maximized. This part is software-specific, so I will focus on how you can do this:

• Steps to graph in MATLAB:

  matlabCopy code
  f = @(x) 1 ./ (2*x.^2 - 3*x + 6);
  fplot(f, [-3, 3])
  grid on

  • Use MATLAB's interactive tools to zoom in and read off the maximum value from the plot.

(b)

Find the derivative of $f(x)$.

Using the quotient rule: $\frac{d}{dx} \left( \frac{1}{2x^2 - 3x + 6} \right) = \frac{-\left( 2x^2 - 3x + 6 \right)'}{(2x^2 - 3x + 6)^2}$ Now, differentiate the numerator $2x^2 - 3x + 6$: $(2x^2 - 3x + 6)' = 4x - 3$ So the derivative is: $f'(x) = \frac{-(4x - 3)}{(2x^2 - 3x + 6)^2}$

(c)

Find the exact value of $x$ where $f(x)$ reaches its maximum.

To find the critical points, set the derivative $f'(x)$ equal to zero: $\frac{-(4x - 3)}{(2x^2 - 3x + 6)^2} = 0$ The numerator must be zero, so: $4x - 3 = 0 \implies x = \frac{3}{4}$

To confirm if this is a maximum, you can take the second derivative (or analyze the behavior of the first derivative around this point). Evaluating the function at $x = \frac{3}{4}$: $f\left( \frac{3}{4} \right) = \frac{1}{2\left( \frac{3}{4} \right)^2 - 3\left( \frac{3}{4} \right) + 6}$ After simplifying: $f\left( \frac{3}{4} \right) = \frac{1}{2 \times \frac{9}{16} - \frac{9}{4} + 6} = \frac{1}{\frac{9}{8} - \frac{18}{8} + \frac{48}{8}} = \frac{1}{\frac{39}{8}} = \frac{8}{39}$

So the function reaches its maximum value at $x = \frac{3}{4}$.

(d)

At what point is the graph of $f(x)$ rising most steeply?

The steepest increase in the function corresponds to the point where the derivative $f'(x)$ is maximized. To do this, we need to find where the absolute value of $f'(x)$ reaches a peak.

Graph the derivative function $f'(x) = \frac{-(4x - 3)}{(2x^2 - 3x + 6)^2}$ in MATLAB using:

matlabCopy code
f_prime = @(x) -(4*x - 3) ./ (2*x.^2 - 3*x + 6).^2;
fplot(f_prime, [-3, 3])
grid on

Look for the point where $f'(x)$ achieves its highest value. This might require numerical methods or a close look at the graph.

Do you want a detailed solution of any part or assistance with MATLAB code?

Here are five related questions to deepen your understanding:

1. How do you apply the quotient rule for more complex rational functions?
2. How do second derivatives help confirm local maxima or minima?
3. What happens to $f(x)$ as $x \to \infty$?
4. Why does the maximum value occur where the derivative is zero?
5. How can we interpret the graph of $f'(x)$ to find points of steepest change?

Tip: Always verify maxima and minima by checking the second derivative or analyzing the behavior of $f'(x)$ around critical points.