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Extrema Problem Solving


If f(x)=ax+bx2+1f(x)=\frac{ax+b}{x^2+1} has a local maximum of 1111 at x=1x=1, what is the value of a+ba+b?

If the polynomial f(x)=x3+3ax2+14bx+cf(x)=x^3+3ax^2+\frac{1}{4}bx+c has local extrema at x=1x=-1 and x=3,x=3, for real numbers aa, bb and cc, what is the value of a+ba+b?

A cubic function f(x)f(x) satisfies the following two conditions:

A.A. f(x)f(x) has an extreme value 4 -4 at x=1  x = 1\ and

B.B. limx0f(x)x=4\displaystyle \lim_{x \to 0} \frac{f(x)}{x} = -4 .

What is the value of f(4) f(4) ?

If the sum of the local extrema of the function f(x)=x33ax2+9x+27 f(x) = x^3 - 3ax^2 + 9x + 27 is 0 0 , what is the value of the real number a a ?

If the slope of the line tangent to the curve y=ax3bx2+cx y = ax^3 - bx^2 + cx at x=2 x = 2 is 15 15 , and (1, 7) (1,\ 7) is the inflection point of the curve, what is the value of a+b+c a + b + c ?


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