Question

use tools use a graphing calculator to estimate the x - coordinates at which the maxima and minima of f(x)=-2x^4 + 5x^3 - 4x^2+3x - 7 occur. round to the nearest hundredth.

Explanation:

Step1: Find the derivative

The derivative of $f(x)=-2x^{4}+5x^{3}-4x^{2}+3x - 7$ is $f'(x)=-8x^{3}+15x^{2}-8x + 3$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$.

Step2: Use a graphing calculator

Input $y = f'(x)=-8x^{3}+15x^{2}-8x + 3$ into a graphing calculator.

Step3: Find the roots

The roots of $f'(x)$ are the $x$ - coordinates of the critical points of $f(x)$. The critical points are where $f'(x)=0$. Use the zero - finding feature of the graphing calculator.

Step4: Classify the critical points

We can use the second - derivative test or analyze the behavior of the function around the critical points. The second - derivative $f''(x)=-24x^{2}+30x - 8$. Evaluate $f''(x)$ at the critical points. If $f''(c)>0$, then $f(x)$ has a local minimum at $x = c$. If $f''(c)<0$, then $f(x)$ has a local maximum at $x = c$.
However, since the problem asks to estimate using a graphing calculator, we can directly observe from the graph of $y = f(x)$ where the maxima and minima occur.
The $x$ - coordinates of the maxima and minima (rounded to the nearest hundredth) are found by using the maximum and minimum features of the graphing calculator on the function $y=-2x^{4}+5x^{3}-4x^{2}+3x - 7$.

Answer:

The $x$ - coordinates of the maxima and minima need to be found using a graphing calculator's maximum and minimum features on the function $y=-2x^{4}+5x^{3}-4x^{2}+3x - 7$. The actual values will depend on the specific graphing calculator used.