## Limits, Continuity and Differentiability Questions and Answers Part-10

1. Let $f\left(x\right)=\begin{cases}x^{2} & if x\leq x_{0}\\ax+b & if x>x_{0} \end{cases}$
The value of the coefficients a and b for which the function is continuous and has a derivative at $x_{0}$ , are
a) $a=x_{0}, b=-x_{0}$
b) $a=2x_{0}, b=-x_0^2$
c) $a=x_0^2, b=-x_{0}$
d) $a=x_{0}, b=-x_0^2$

Explanation: For f to be continuous everywhere, we must have

2. Given f' (2) = 6 and f' (1) = 4, $\lim_{h \rightarrow 0}\frac{f\left(2h+2+h^{2}\right)-f\left(2\right)}{f\left(h-h^{2}+1\right)-f\left(1\right)}$
is equal to
a) 3/2
b) 3
c) 5/2
d) -3

Explanation:

3. Let $f:R\rightarrow R$   be such that f (1) = 3 and f' (1) = 6. Then $\lim_{x \rightarrow 0}\left(\frac{f\left(1+x\right)}{f\left(1\right)}\right)^{1/x}$     equals
a) 1
b) $e^{1/2}$
c) $e^{2}$
d) $e^{3}$

Explanation:

4.The domain of the derivative of the function $f\left(x\right)=\begin{cases}\tan^{-1}x &\mid x\mid \leq1\\\frac{1}{2}\left(\mid x\mid-1\right) & \mid x\mid > 1\end{cases}$
a) $R \sim\left\{0\right\}$
b) $R \sim\left\{1\right\}$
c) $R \sim\left\{-1\right\}$
d) $R \sim\left\{-1,1\right\}$

Explanation:

5. If f (1) = 1, f' (1) = 3 then the derivative of y = f (f ( f ( f (x))) at x = 1 is
a) 256
b) 16
c) 81
d) 27

Explanation: y'(x) = f'(f(f(f(x))) f'(f(f(x))) f'( f(x)) f'(x)

6. $f\left(x\right)=\begin{cases}ax & x < 2\\ax^{2}-bx+3 & x \geq 2\end{cases}$
If f is differentiable for all x then
a) a = 3/4, b = 9/4
b) a = 1, b = 2
c) a = 3/2, b = 9/2
d) a = 3/4, b = 9/2

Explanation: Since f is differentiable for all x, in particular

7. Let f and g be differentiable function such that f' (x) = 2g(x) and g' (x) = – f (x), and let $T\left(x\right)=\left(f\left(x\right)\right)^{2}-\left(g\left(x\right)\right)^{2}.$
Then T' (x) is equal to
a) T(x)
b) 0
c) 2f (x) g(x)
d) 6 f (x) g(x)

Explanation:

8. Let f be a twice differentiable function such that f" (x) = – f (x) and f' (x) = g(x). If $h'\left(x\right)=\left[f\left(x\right)\right]^{2}+\left[g\left(x\right)\right]^{2}$      , h(1) = 7 and h(0) = 2, then h(3) is equal to
a) 11
b) 4
c) 14
d) 13

Explanation:

9. If $y^{2}=P\left(x\right)$   is a polynomial of degree 3,
$2\frac{d}{dx}\left(y^{3}\frac{d^{2}y}{dx^{2}}\right)$     is equal to
a) P(x) + P'(x)
b) P(x) P'(x)
c) P(x) P'''(x)
d) a constant

10. Let $f\left(x\right)=\lim_{n \rightarrow \infty}\frac{\left(x^{2}+x+\frac{9}{4}+\cos\pi x\right)^{n}-1}{\left(x^{2}+x+\frac{9}{4}+\cos\pi x\right)^{n}+1}.\left(\frac{2\tan ^{n}x+x^{n}}{\tan^{n}x}\right)$
a) f is differentiable for all $x\epsilon R$
c) f is discontinuous at all $n\epsilon I$