Consider the fourth-degree polynomial f(x) = x4 + ax2 + b. (i) Show that there is one critical number when a = 0. Then find the open intervals on which the function is increasing or decreasing. (ii) Show that there is one critical number when a > 0. Then find the open intervals on which the function is increasing or decreasing. (iii) Show that there are three critical numbers when a < 0. Then find the open intervals on which the function is increasing or decreasing. (iv) Show that there are no real zeros when a2 < 4b. (v) Determine the possible number of zeros when a2 ≥ 4b. Explain your reasoning.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Consider the fourth-degree polynomial f(x) = x4 + ax2 + b. (i) Show that there is one critical number when a = 0. Then find the open intervals on which the function is increasing or decreasing. (ii) Show that there is one critical number when a > 0. Then find the open intervals on which the function is increasing or decreasing. (iii) Show that there are three critical numbers when a < 0. Then find the open intervals on which the function is increasing or decreasing. (iv) Show that there are no real zeros when a2 < 4b. (v) Determine the possible number of zeros when a2 ≥ 4b. Explain your reasoning.
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