Let `f(x)=a x^2-b|x|,` where `aa n db` are constants. Then at `x=0,f(x)` has a maxima whenever `a >0,b >0` a maxima whenever `a >0,b<0` minima whenever `a >0,b<0` neither a maxima nor a minima whenever `a >0, b<0`

Let f(x)=a sin x+c , where a and c are real numbers and a>0. Then f(x)lt0, AA x in R if

Let f(x)=a x^2+b x+c. Consider the following diagram. Then Fig c 0 a+b-c >0 a b c<0

Let f(x) = ax^(2) - bx + c^(2), b ne 0 and f(x) ne 0 for all x in R . Then

Let f(x)=a^2+b x+c where a ,b , c in Ra n da!=0. It is known that f(5)=-3f(2) and that 3 is a root of f(x)=0. then find the other of f(x)=0.

Let f(x)= {x^(3)+x^(2)+10x, xlt0 .Investigate x=0 for local -3sinx,xge0 maxima/minima.

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a, x=b, y=f(x) and y=f(c)("where "c in (a,b))" is minimum when "c=(a+b)/(2) . "Proof : " A=int_(a)^(c)(f(c)-f(x))dx+int_(c)^(b)(f(c))dx =f(c)(c-a)-int_(a)^(c)(f(x))dx+int_(a)^(b)(f(x))dx-f(c)(b-c) rArr" "A=[2c-(a+b)]f(c)+int_(c)^(b)(f(x))dx-int_(a)^(c)(f(x))dx Differentiating w.r.t. c, we get (dA)/(dc)=[2c-(a+b)]f'(c)+2f(c)+0-f(c)-(f(c)-0) For maxima and minima , (dA)/(dc)=0 rArr" "f'(c)[2c-(a+b)]=0(as f'(c)ne 0) Hence, c=(a+b)/(2) "Also for "clt(a+b)/(2),(dA)/(dc)lt0" and for "cgt(a+b)/(2),(dA)/(dc)gt0 Hence, A is minimum when c=(a+b)/(2) . If the area bounded by f(x)=(x^(3))/(3)-x^(2)+a and the straight lines x=0, x=2, and the x-axis is minimum, then the value of a is

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a, x=b, y=f(x) and y=f(c)("where "c in (a,b))" is minimum when "c=(a+b)/(2) . "Proof : " A=int_(a)^(c)(f(c)-f(x))dx+int_(c)^(b)(f(c))dx =f(c)(c-a)-int_(a)^(c) (f(x))dx+int_(a)^(b)(f(x))dx-f(c)(b-c) rArr" "A=[2c-(a+b)]f(c)+int_(c)^(b)(f(x))dx-int_(a)^(c)(f(x))dx Differentiating w.r.t. c, we get (dA)/(dc)=[2c-(a+b)]f'(c)+2f(c)+0-f(c)-(f(c)-0) For maxima and minima , (dA)/(dc)=0 rArr" "f'(c)[2c-(a+b)]=0(as f'(c)ne 0) Hence, c=(a+b)/(2) "Also for "clt(a+b)/(2),(dA)/(dc)lt0" and for "cgt(a+b)/(2),(dA)/(dc)gt0 Hence, A is minimum when c=(a+b)/(2) . If the area enclosed by f(x)= sin x + cos x, y=a between two consecutive points of extremum is minimum, then the value of a is

Let f(x)={(1+3x)^(1/x), x != 0e^3,x=0 Discuss the continuity of f(x) at (a) x=0, (b) x=1

If f(x)=x^3+b x^2+c x+da n df(0),f(-1) are odd integers, prove that f(x)=0 cannot have all integral roots.

If b gt a , then the equation (x-a)(x-b)-1=0 , has