Question

Describe the behavior of the graph at the x-intercepts for the function f(x)=(2x-7)^7(x+3)^4. Identify each x-intercept and justify your answer as well.

Answer 1

Find $x$-intercepts

$f(x)=0 \\(2x-7)^7(x+3)^4=0 \\2x-7=0 \\x=3.5 \\x+3=0 \\x=-3$

Determine the sign of the function:
x∈(-∞,-3)∪(-3,3.5)⇒y<0
x∈(3,5,+∞)⇒y>0

Therefore, point $x=-3$ is the maximum and point $x=3.5$ is the point of inflection.