change of graph of f(x)=x^3+kx when k is <, >, or = to

[smileykd]

Let f(x) = x^3 + kx, where k is a constant. Discuss the number of local extrema and the shape of the graph of f if k>0, k<0, and k=0.
I plugged these in my calculator and saw that the graph changes when you plug in different constants, but I don't know why. Can anyone explain it please?

 

[tkhunny]

You're in calculus, right?

f'(x) = 3*x^2 + k

What does 'k' do to that?

 

[smileykd]

When I graph them, for k>0 and k=0 I am getting a parabola, but for k<0, I am getting a graph of a fourth degree polynomial with two minimums and one maximum. That's what I'm not understanding.

 

[skeeter]

When I graph them, for k>0 and k=0 I am getting a parabola, but for k<0, I am getting a graph of a fourth degree polynomial with two minimums and one maximum. That's what I'm not understanding.

huh?

f'(x) = 3x² + k

if k > 0, f'(x) never intersects the x-axis
if k = 0, f'(x) "touches" the x-axis at a single point
if k < 0, then you get f'(x) to intersect the x-axis twice

what does all this information about f'(x) tell you about f(x)?

 

[tkhunny]

When I graph them, for k>0 and k=0 I am getting a parabola, but for k<0, I am getting a graph of a fourth degree polynomial with two minimums and one maximum. That's what I'm not understanding.

That's easy. You are simply doing something wrong. You have it coded incorrectly in your calculator, maybe? Changing the value of 'k' should not affect the degree of the polynomial. Definitely something wrong with that.