A graphing calculator is recommended.
Graph the polynomial, and determine how many local maxima and minima it has.

The polynomial has —Select— and – Select— Watch It
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Ask by Peterson Dickson. in the United States

Mar 28,2025

Question

A graphing calculator is recommended.
Graph the polynomial, and determine how many local maxima and minima it has.

The polynomial has —Select— and – Select— Watch It
Need Help?

Ask by Peterson Dickson. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

**Step 1. Find the derivative of $$ y $$.**

Given
$$
y = 5x^3 - 5x^2 - 3x,
$$
differentiate with respect to $$ x $$:
$$
y' = \frac{d}{dx}(5x^3) - \frac{d}{dx}(5x^2) - \frac{d}{dx}(3x) = 15x^2 - 10x - 3.
$$

**Step 2. Set the derivative equal to zero to find critical points.**

Solve
$$
15x^2 - 10x - 3 = 0.
$$

Use the quadratic formula:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$
where $$ a = 15 $$, $$ b = -10 $$, and $$ c = -3 $$. Substitute these in:
$$
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(15)(-3)}}{2(15)} = \frac{10 \pm \sqrt{100 + 180}}{30} = \frac{10 \pm \sqrt{280}}{30}.
$$

Simplify the radical:
$$
\sqrt{280} = \sqrt{4 \cdot 70} = 2\sqrt{70}.
$$

Thus,
$$
x = \frac{10 \pm 2\sqrt{70}}{30} = \frac{5 \pm \sqrt{70}}{15}.
$$

**Step 3. Determine the nature of the critical points.**

Compute the second derivative:
$$
y'' = \frac{d}{dx}(15x^2 - 10x - 3) = 30x - 10.
$$

Evaluate $$ y'' $$ at each critical point.

- For
  $$
  x = \frac{5 + \sqrt{70}}{15},
  $$
  approximate value: since $$ \sqrt{70} \approx 8.3666 $$, 
  $$
  x \approx \frac{5 + 8.3666}{15} \approx 0.8911.
  $$
  Then:
  $$
  y''(0.8911) = 30(0.8911) - 10 \approx 26.73 - 10 = 16.73 > 0.
  $$
  So, this critical point is a **local minimum**.

- For
  $$
  x = \frac{5 - \sqrt{70}}{15},
  $$
  approximate value:
  $$
  x \approx \frac{5 - 8.3666}{15} \approx -0.22444.
  $$
  Then:
  $$
  y''(-0.22444) = 30(-0.22444) - 10 \approx -6.7333 - 10 = -16.73 < 0.
  $$
  So, this critical point is a **local maximum**.

**Step 4. Conclusion**

The polynomial
$$
y = 5x^3 - 5x^2 - 3x
$$
has one local maximum and one local minimum.
The polynomial $$ y = 5x^3 - 5x^2 - 3x $$ has **one local maximum** and **one local minimum**.

A graphing calculator is recommended.
Graph the polynomial, and determine how many local maxima and minima it has.

The polynomial has —Select— and – Select— Watch It
Need Help?

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A graphing calculator is recommended. Graph the polynomial, and determine how many local maxima and minima it has. The polynomial has —Select— and – Select— Watch It Need Help?

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A graphing calculator is recommended.
Graph the polynomial, and determine how many local maxima and minima it has.

The polynomial has —Select— and – Select— Watch It
Need Help?