Show, using only calculus, that f(x) = x2 - 2x + 3 takes on only positive values

Question

Show, using only calculus, that f(x) = x2 - 2x + 3 takes on only positive values. Do NOT claim that f(x) is always positive because that's what the graph looks like! (You may want to consider the minimum value of f(x).) Let g(x) = x2 + rx + s. Using only calculus, find a condition on the values of r and s so that g(x) takes on only positive values. A quadratic y = ax2 + bx + c crosses the x-axis when the equation 0 = ax2 + bx + c has at least one solution. Recall that if there are solutions, they have the form If the quadratic g(x) as in part (b) takes on only positive values, this means that it never crosses the x-axis. Relate the condition you found in (b) to the information above. How might the condition from (b) also tell you that g(x) never crosses the x-axis? Give 2 examples of pairs of r and s for which g takes on both positive and negative values.

Explanation / Answer

r = -4 , s= 2

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