Thread: Graphing polynomial fucntions

For the function g(x)=(1-2x)^2(x-3) find
a) the x and y intercepts
b) the first and second derivatives,
c)the intervals on which the function is increasing and decreasing,
d) inflection points and concavity.
e) finally sketch the function

For a) I got a y intercept of (0,-3) and x-intercepts of (3,0) and (1/2,0).
b)For the first derivative I got
f'(x)= -4(1-2x)(x-3) + (1-2x)^2
f''(x)= 12x^2-32x+ 13
Can somebody please confirm whether the above two answers are correct please?

The problem I'm having is with the question c) and d). How do I figure out what the intervals are in order to test where the function is increasing or decreasing?

Originally Posted by anne25

b)For the first derivative I got
f'(x)= -4(1-2x)(x-3) + (1-2x)^2
f''(x)= 12x^2-32x+ 13

The first derivative is correct. Since it is a second-degree polynomial, the second derivative must be a first-degree polynomial.

Originally Posted by anne25

The problem I'm having is with the question c) and d). How do I figure out what the intervals are in order to test where the function is increasing or decreasing?

f(x) is increasing when f'(x) > 0 and it is decreasing when f(x) < 0. You need to solve a quadratic inequality. Similarly, f(x) is convex when f''(x) > 0 and f(x) is concave when f''(x) < 0.