Necessary Condition for an Extrema(Is a Function Differentiable at Boundaries)

Q. Let $\sum _{k=1}^{10}f(a+k)=16(2^{10}-1)$, where the function $f$ satisfies $f(x+y)=f\left(x\right)+f\left(y\right)$ for all natural numbers $x,y$ and $f\left(1\right)=2$. Then the natural number '$a$' is:

1. $3$
2. $2$
3. $16$
4. $4$

Q. The necessary condition to be maximum or minimum for the function is

1. f"(x) = 0 and it is sufficient

2. f'(x) = 0 and f"(x) = -ve
3. f(x) = 0 and it is sufficient

4. f'(x) = 0 but it is not sufficient

Q.

Suppose f(x) = and iff(x) = f(1) what are possible values of a and b?

Q.

Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

Q. Let $f:\to$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right),\forall x,y$ . If $f\left(x\right)$ is differentiable at $x=0$, then

1. $f\left(x\right)$ is continuous $\forall x$
2. $f^{\prime }\left(x\right)$ is continuous $\forall x$
3. $f\left(x\right)$ is differentiable only in a finite interval containing zero
4. $f\left(x\right)$ is is differentiable except at finitely many points

Q. If $0<x<1$ and $y=\frac{1}{2}{x}^2+\frac{2}{3}{x}^3+\frac{3}{4}{x}^4+\cdots ,$ then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is

1. $2e$
2. $\frac{1}{2}{e}^2$
3. $2e^2$
4. $\frac{1}{2}\sqrt{e}$

Q. Suppose $f\left(x\right)=x^3+a{x}^2+bx+c$ satisfies $f(–2)=-10$ and takes the extreme value $\frac{50}{27}$ at $x=\frac{2}{3}.$ Then the value of $a+b+c$ is

Q. If $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{1+e^{1/x}}& ,x\ne 0\\ 0& ,x=0\end{array}\right.$ then f (x) is

(a) continuous as well as differentiable at x = 0
(b) continuous but not differentiable at x = 0
(c) differentiable but not continuous at x = 0
(d) none of these

Q.

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = (2x − 1)² + 3 (ii) f(x) = 9x² + 12x + 2

(iii) f(x) = −(x − 1)² + 10 (iv) g(x) = x³ + 1

Q.

If f(x) =.

For what value (s) of a does f(x) exists?

Q.

For f function y = f(x) if we have f^'(c) = 0 and f^" > 0 then x = c is a point of

1. point of inflexion

2. absolute minima

3. local maxima

4. local minima

Q.

If f (x) = a log |x| + bx² + x has extreme values at x = –1 and at x = 2, then values of a and b are

1. 2, -1/4

2. 2, -1/2

3. 2, -1

4. 2, -5/6

Q.

Find f(x) andf(x), where f(x) =

Q.

Find all the points of discontinuity of f(x) defined by f(x)=|x|-|x+1|.

Q. The least integral value in range of f(x) = 3x + 3–x is

Q.

If f(x) and f’(x) are differentiable at x = c, then the necessary condition for f(c) to be an extremum of f(x) is -

1. f’(c) = 0

2. f(c) = 0

3. f”(c) = 0

4. None of these

Q.

Evaluatef(x), where f(x) =

Q. The set of points where $f\left(x\right)=x^2|x|$ is thrice differentiable, is:

1. $R-\left\{0\right\}$
2. $R-\left\{1\right\}$
3. $R$
4. $R-\{-1\}$

Q. If for a function f(x), f(a) = 0, f'(a)=0, f"(a) > 0, then at x = a, f(x) is
[MP PET 1994; Pb. CET 2002]

1. Minimum

2. Maximum

3. Not an extreme point

4. Extreme point

Q. The value of c in Rolle's theorem for the function f (x) = x³ − 3x in the interval [0, $\sqrt{3}$] is
(a) 1
(b) −1
(c) 3/2
(d) 1/3

Q. Write necessary condition for a point x = c to be an extreme point of the function f(x).

Q.

Is the function defined by continuous at x = $\mathrm{\pi }$?

Q. If for a function f(x), f(a) = 0, f'(a)=0, f"(a) > 0, then at x = a, f(x) is
[MP PET 1994; Pb. CET 2002]

1. Minimum

2. Not an extreme point

3. Extreme point

4. Maximum

Q. the local minimum value of the function f given by f(x) = 3+lxl is what?

Q.

Find f(x), where f(x) =

Q.

1. Second order, second degree equation

2. Third order, third degree equation

3. Second order, first degree equation

4. Second order, third degree equation

Q.

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = (2x − 1)² + 3 (ii) f(x) = 9x² + 12x + 2

(iii) f(x) = −(x − 1)² + 10 (iv) g(x) = x³ + 1

Q.

Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

Q. Give an example of a function which is continuos but not differentiable at at a point.

Q.

If f(x) and f’(x) are differentiable at x = c, then the necessary condition for f(c) to be an extremum of f(x) is -

1. f(c) = 0

2. f’(c) = 0

3. f”(c) = 0

4. None of these