Let 𝑦=−3𝑥4+6𝑥+5
(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.
(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).