15. If f(x)=x^3 , then lim (f(2+h)-f(2))/(h)=underline ( ) A. 4 B. -4 C. 12 D. infty 16. Let f(x)x^3-12x+1 be defined on [0,3] What is the range of f? A. (-infty ,infty ) B. (-infty ,1] C. [-15,1] D. [-15,17) 17. If f(x)=sqrt (2+sqrt (x)) then f^1(x)=underline ( ) A. (1)/(sqrt (2x+sqrt (x)+)) B. (1)/(4sqrt (2x+xsqrt (x))) (x)/(1+x) D. 2x 18. Which of the ff is the set of all critical numbers of f(x)=. f(x)= ) x-1,xlt 1 x^2-x,xgeqslant 1 A. 1 B. (1)/(2) C. (1)/(2),1 D. 19. int _((pi )/(4))^pi /3(1)/(x^2) dx=underline ( ) A. -pi /12 B. -12pi C. -pi D. -1 20. If the sum of n-terms of an arithmetic sequence is 3n^2+5n and a_(n)=164 then what is the nu terms? A. 21 C. 45 B. 27 D. 71 21. What are the respective valued of A ,B and C so that (x-1)/(x^3)+x=(A)/(x)+(Bx+C)/(x^2)+1 A. -1 , 1,1 B. -1 , 1,2 C. 1. -2 o D. 3.1.2 22. Which of the ff is not true about the graph of g(x)=(x^2-1)/(x^2)+1 A. Range of g is (-infty ,1) C. y=1 is H.A B. g is an even function D. Asx-infty ,g(x)-1 23. Let f(x)=x^4-6x^2+1 on [-1,3) The local maximum value of f is __ A f(0) B f(sqrt (3)) C f(3) D f(-sqrt (3))

## Step 1: Evaluating the limit expression### The question asks for the value of , where . We substitute into the expression: . Expanding gives . Substituting this back into the limit expression, we get . As approaches 0, the expression approaches 12.## Step 2: Determining the range of f(x)### The function is defined on the closed interval . We check the endpoints and critical points to find the range. . . To find critical points, we take the derivative: . Setting , we get , so , and . Since we are considering the interval , we only consider . . The minimum value is -15 and the maximum is 1. Thus, the range is .## Step 3: Finding the inverse function### Given , we want to find . Let , so . Squaring both sides gives . Then, , and squaring again gives . Thus, . The question asks for the derivative of the inverse, which isn't provided in the options.## Step 4: Identifying critical numbers### Critical numbers occur where the derivative is zero or undefined. For , , so . For , , so . Setting , we get . However, this is not in the domain . The derivative is undefined at due to the piecewise definition. Checking the limit from the left, we have 1, and from the right, we have . The derivative exists at . Thus, there are no critical numbers.## Step 5: Evaluating the definite integral### We evaluate .## Step 6: Finding the number of terms### The sum of an arithmetic sequence is given by . We know that . The general term of an arithmetic sequence is . The sum can also be expressed as . We have . . , so . Then , so , and , thus .## Step 7: Partial fraction decomposition### We have . Multiplying by , we get . If , . Then , so . Thus, and , so . Therefore, , , and .## Step 8: Analyzing the graph of g(x)### The function . As , , so is a horizontal asymptote. Since , the function is even. The range is , as the numerator is always less than the denominator.## Step 9: Finding the local maximum### . . Critical points are and . Since the interval is , we consider and . . . . . The local maximum on is , but the maximum value is .