Let 𝑦=−3𝑥4+6𝑥+5

Question

(a) Calculate 𝑦′(1) and 𝑦″(1)
(b) Which of the following statements is true?

Select one:

a) Near 𝑥=1, the graph of 𝑦 is decreasing at a decreasing rate.

b) Near 𝑥=1, the graph of 𝑦 is increasing at an increasing rate.

c) Near 𝑥=1, the graph of 𝑦 is increasing at a decreasing rate.

d)Near 𝑥=1, the graph of 𝑦 is decreasing at an increasing rate.

Answer 1

(a) 𝑦′(𝑥) = −12𝑥³+6, so 𝑦′(1) = −12(1)³+6 = −6

𝑦″(𝑥) = −36𝑥², so 𝑦″(1) = −36(1)²=−36

(b) d) Near 𝑥=1, the graph of 𝑦 is decreasing at an increasing rate. This is because 𝑦′(1) is negative (meaning the graph is decreasing) and 𝑦″(1) is also negative (meaning the rate of decrease is increasing).