Consider $$f(x)=x3+ax2+bx+c$$. Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that $$f(x)$$ is an increasing function is

Question

Consider $$f(x)=x3+ax2+bx+c$$. Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that $$f(x)$$ is an increasing function is

Infinity Learn · Accepted Answer

f'(x)=3x2+2ax+by$$ $$=$$ $$f(x) is increasing⇒ f'(x)≥$$0$$,∀x and for f'(x)=0$$ should not form an interval⇒ $$(2a)2$$−$$4$$×$$3$$×b≤$$0$$⇒$$a2$$−$$3b$$≤$$0This$$ is true for exactly $$16$$ ordered pairs $$(a$$, $$b), $$1$$≤a, b≤$$6$$, namely (1$$, $$1), (1$$, $$2), (1$$, $$3), (1$$, $$4), (1$$, $$5), (1$$, $$6), (2$$, $$2), (2$$, $$3), (2$$, $$4), (2$$, $$5), (2$$, $$6), (3$$, $$3), (3$$, $$4), (3$$, $$5), (3$$, $$6), and (4$$,$$6). Thus, the required probability is $$16/36$$ $$=$$ $$4/9$$

Consider $f (x) = x^{3} + {ax}^{2} + bx + c$ . Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that f(x) is an increasing function is

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Consider f(x)=x3+ax2+bx+c. Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that f(x) is an increasing function is

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Consider $f (x) = x^{3} + {ax}^{2} + bx + c$ . Parameters a, b, c are chosen, respectively, by throwing a die three times. Then the probability that f(x) is an increasing function is