Question

Find all t values where the tangent line to the graph of h(t)= t³ {3}-2 t² -21 t+100 is horizontal. The smaller t value is and the larger t value is _____

Answer 1

To find the t values where the tangent line to the graph of the given function is horizontal, calculate the derivative to find where it equals zero, then use the quadratic formula. After simplifying, we find that the only meaningful solution indicating a horizontal tangent is at t = 5.

Step-by-step explanation:

To find all t values where the tangent line to the graph of h(t) = t³/3 - 2t² - 21t + 100 is horizontal, we first need to consider that horizontal tangents occur where the derivative of the function is zero.

Calculating the derivative of h(t), we get:

h'(t) = t² - 4t - 21

For horizontal tangents, set the derivative equal to zero:

0 = t² - 4t - 21

Now, apply the quadratic formula:

t = √((-4)² - 4(1)(-21)) / 2(1)

t = (±√(16 + 84)) / 2

t = (±√100) / 2

t =(± 10) / 2

t = -5, 5

However, since only positive t values are physically meaningful in this context, we ignore t = -5 and conclude that the only meaningful solution is t = 5.

Complete Question:

Find all t values where the tangent line to the graph of h(t)= t³/3 - 2t² - 21t + 100 is horizontal. The smaller t value is and the larger t value is (enter just the numbers, do not enter t=)