Hard Question HELPP

Question

Construct a function that has the same slope at x=1 and x=2. Then find two points where y=x*4-2x*2 has the same tangent line.(draw the graph)

Answer 1

Answer: y=-1 and 2 points on the graph with same tangent slope are (-1,-1) and (1,-1)

x⁴-2x² = x²(x²-2) = x(x)(x+square root of 2)(x-square root of 2) This function is degree 4 and has potentially 4 zeroes, or 4 places where it intersects the x axis. Factor it, set each factor = 0, solve for x, and the zeroes are x=0 and x=+ or - square root of 2. The graph intersects the x axis twice at 0,+2^1/2 and -2^1/2. (at 0, it just touches, at + and -2^1/2 it crosses the x-axis) Or you could say it repeats 2 of its zeroes, so you get 4, as 0,0,+ and -2^1/2.

The slope of the graph is the first derivative or 4x³-4x or 4x(x²-1) or 4x(x+1)(x-1) This has slope = 0 at x=0 and x=1 and x=-1. It has 3 points with slope equal to zero. These are local minimums and a local maximum in the graph.

Just set each factor = 0 and solve for x

At x=0 the slope is 0, at x= plus or minus 1, the slope is 0, at x=2, the slope is 4(8)-4(2)=24. After x=1, the slope increases exponentially. To the left of x=-1, the slope decreases exponentially, getting more & more negative.

A function that has the same slope at x=1 and x=2 is y=any constant, such as y=-1

This 4 degree function has 3 changes in direction, first down, then up briefly, then down briefly, then up forever. On the left the graph comes down from infinity. We know the 4th powered, highest degree term dominates at high x values or low x values, very positive or very negative values. In between the graph after crossing the x axis at about x=-1.414, then reaches bottom at x=-1, (-1,-1)a local and global minimum, then turns upwards reaching a local maximum at x=0, at the origin. Then it changes direction again, moving downward briefly, until x=1 reaching another local & global minimum,(1,-1) then changing direction going upwards, crossing the x axis at x=about 1.414, then continuing upwards forever.

The function y=0 has the same slope at x=1 and x=2. It's a flat straight horizontal line with slope=0

Two points on the 4 degree function that have a tangent line with the same slope of 0 are x=-1 and x=1

The function y=-1 is tangent to the 4 degree function at x=-1 and x=1

y=-1 at x=-1 and y=-1 at x=1 for both y=-1 and for the 4 degree function.

The global maximum of the graph approaches infinity. But a local maximum is at x=0 y=0, (0,0) or the origin, half way between the two x values of the two local (& global) minimums (-1,-1), (1,-1)