Question 1a For each of the following statements, determine whether it is true o

Question

Question 1a

For each of the following statements, determine whether it is true or false and justify your answer. If the differentiable function f: R rightarrow R is strictly increasing, then f'(x)>0 for all x. If the differentiable function f: R rightarrow R is monotonically increasing then f'(x) ge 0 for all x. If the function f : R rightarrow R is differentiable and f(x) le f(0) for all x in [-1, 1], then f'(0)=0. If the function f: R rightarrow R is differentiable and f(x) le f(1) for all x in {-1, 1], then f'(1)=0. Sketch the graphs of the following functions. Find the intervals on which they are increasing or decreasing. f:R rightarrow R defined by f(x)=x3 + ax2 + bx + c for all x. h : (0, infinity) rightarrow R defined by h(x)=a + b/x for x > 0, where a > 0,b > 0. For real numbers a, b, c, and d, define O={x|cx+d 0}. Then define f(x)=ax + b/cx + d for all x in O. Show that if the function f : O rightarrow R is not constant, then it fails to have any local maximizers or minimizers. Sketch the graph. For c>0, prove that the following equation does not have two solutions : x3 - 3x + c=0, 0

Explanation / Answer

question 1- a is true

as the criteria for monotonic function is f'(x) > 0

thus

because f'(x)= lim h--->0 [f(x+h)-f(x)]h

since the function is increasing so [f(x+h)>f(x)

thus f'(x)>0

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