habits of mind use appropriate tools estimate the turning point of the graph of p(x)= - 3x^4 + 4x^3 - 2x^2 +…

Explanation:

Step1: Find the derivative

The derivative of (p(x)=- 3x^{4}+4x^{3}-2x^{2}+5x + 10) using the power - rule ((x^n)^\prime=nx^{n - 1}) is (p^\prime(x)=-12x^{3}+12x^{2}-4x + 5).

Step2: Set the derivative equal to zero

To find the turning points, we set (p^\prime(x) = 0), i.e., (-12x^{3}+12x^{2}-4x + 5=0). This is a cubic equation. We can use numerical methods (such as Newton - Raphson method) or a graphing utility to approximate the roots. Using a graphing calculator or software, we find the approximate value of (x) for which (p^\prime(x)=0).

Step3: Analyze the meaning

The turning point of the graph of (y = p(x)) represents a point where the function changes from increasing to decreasing or vice - versa. In the context of "Try It! 5", it could represent a maximum or minimum value (local or global depending on the domain) of the quantity represented by the function (p(x)), or a point where the rate of change of the quantity changes sign.

Answer:

The turning point is found by solving (-12x^{3}+12x^{2}-4x + 5 = 0) approximately using numerical methods or a graphing utility. It represents a point where the function (p(x)) changes from increasing to decreasing or vice - versa in the context of "Try It! 5".