The number of turning points in the graph of a polynomial function is equal to the number of real roots of its derivative.
First, let's find the derivative of m(x) using the product rule and the chain rule:
m'(x) = (2x + 5)(3(x - 3)^2)(8x + 1)^5 + (2(x - 3)^3)(8x + 1)^5 + (2x + 5)(x - 3)^3(5(8x + 1)^4)(8)
Simplifying this expression:
m'(x) = (3(2x + 5)(x - 3)^2)(8x + 1)^5 + (2(x - 3)^3)(8x + 1)^5 + (10(2x + 5)(x - 3)^3)(8x + 1)^4
Now, to find the number of real roots, we need to determine the number of sign changes in the coefficients of m'(x).
Let's examine the signs of each term of m'(x) as x approaches positive and negative infinity.
As x approaches positive infinity:
- The first term (3(2x + 5)(x - 3)^2)(8x + 1)^5) is positive.
- The second term (2(x - 3)^3)(8x + 1)^5) is negative.
- The third term (10(2x + 5)(x - 3)^3)(8x + 1)^4) is positive.
As x approaches negative infinity:
- The first term (3(2x + 5)(x - 3)^2)(8x + 1)^5) is negative.
- The second term (2(x - 3)^3)(8x + 1)^5) is negative.
- The third term (10(2x + 5)(x - 3)^3)(8x + 1)^4) is positive.
There is one sign change from positive to negative and one sign change from negative to positive. Therefore, the number of real roots of m'(x) is 2.
So, the graph of m(x) should have 2 turning points.
Brandy is asked to sketch a graph of, m(x) = (2x + 5)(x - 3)^3(8x + 1)^5. How many turning points should her sketch include?
1 answer