Examine the differentiability of f x = 1 - e - x 2 at x = 0 .

Given, f x = 1 - e - x 2 at x = 0 Here, f x = 0 Now, LHD = f ' 0 - = lim h → 0 f 0 - h - f 0 - h = lim h → 0 1 - e - h 2 - h = lim h → 0 - 1 e h 2 e h 2 - 1 h 2 = lim h → 0 - 1 e h 2 × lim h → 0 e h 2 - 1 h 2 = - 1 , lim x → 0 e x - 1 x = 1 Similarly, RHD = f ' 0 + = lim h → 0 f 0 + h - f 0 h = lim h → 0 1 - e - h 2 h = lim h → 0 1 e h 2 e h 2 - 1 h 2 = 1 Since, LHD ≠ RHD f x is not differentiable at x = 0

If f x = a x 2 - b if x < 1 - 1 x if x ≥ 1 is derivable at x = 1 . Find the values of a and b .

Given, f x = a x 2 - b if x < 1 - 1 x if x ≥ 1 is derivable at x = 1 We observe that f x = 1 x if x ≤ - 1 a x 2 - b if - 1 < x < 1 - 1 x if x ≥ 1 Therefore, at x = 1 LHL = lim x → 1 - f x = lim x → 1 - a x 2 - b = a - b RHL = lim x → 1 + f x = lim x → 1 + - 1 x = - 1 f 1 = - 1 Since, f x is differentiable at x = 1 Therefore, it is also continuous at x = 1 So, we say that lim x → 1 - f x = lim x → 1 + f x = f 1 Therefore, a - b = - 1 . . . ( i ) Now differentiating above function, f ' x = - 1 x 2 if x ≤ - 1 2 a x if - 1 < x < 1 1 x 2 if x ≥ 1 As it is differentiable at x = 1 Therefore, LHD at x → 1 - = 2 a RHD at x → 1 + = 1 As, LHD = RHD Therefore, 2 a = 1 , a = 1 2 Now, b = a + 1 , b = 3 2 using i

Discuss the continuity & derivability of f x = x - 1 2 ; 0 ≤ x < 1 x x ; 1 ≤ x ≤ 2 , where x indicates the greatest integer x

We have, f x = x - 1 2 ; 0 ≤ x < 1 x x ; 1 ≤ x ≤ 2 ⇒ f x = 1 2 - x ; 0 ≤ x < 1 2 x - 1 2 ; 1 2 ≤ x < 1 x ; 1 ≤ x < 2 2 x ; x = 2 Now, at x = 1 2 ⇒ f 1 2 = 0 LHL = lim x → 1 - 2 f x = lim x → 1 - 2 1 2 - x = 0 RHL = lim x → 1 2 + f x = lim x → 1 2 + x - 1 2 = 0 Therefore, f x is continuous at x = 1 2 Similarly, at x = 1 , f 1 = 1 LHL = lim x → 1 - f x = lim x → 1 - x - 1 2 = 1 2 RHL = lim x → 1 + f x = lim x → 1 + x = 1 Since, LHL ≠ RHL , therefore f x is neither continuous nor differentiable at x = 1 Now, LHL = lim x → 2 - f x = lim x → 2 - x = 2 RHL = lim x → 2 + f x = lim x → 2 + 2 x = 4 Since, LHL ≠ RHL , therefore f x is neither continuous nor differentiable at x = 2 . Also, at x = 1 2 f ' x = - 1 ; 0 < x < 1 2 1 ; 1 2 < x < 1 f ' 1 2 - = - 1 and f ' 1 2 + = 1 Since, LHD ≠ RHD , therefore f x is not differentiable at x = 1 Ans. f x is continuous but not differentiable at x = 1 2 and neither continuous nor differentiable at x = 1 , 2 .

Let f : [ a , b ] → [ 1 , ∞ ) be a continuous function and let g : R → R be defined as g x = 0 i f x < 0 ∫ a x f t d t i f a ≤ x ≤ b ∫ a b f t d t i f x > b . Then :

g x is continuous but not differentiable at b

g x is continuous and differentiable at either a or b but not both

For every twice differentiable function f : ℝ → - 2 , 2 with f 0 2 + f ′ 0 2 = 85 , which of the following statement(s) is (are) TRUE?

There exists α ∈ - 4 , 4 such that f α + f ′ ′ α = 0 a n d f ′ α ≠ 0

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 35

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IMPORTANT

Earn 100

If $g (x) = (|x - 1| + |4 x - 11|) [x^{2} - 2 x - 2],$ then find the number of point of discontinuity of $g (x)$ in $(\frac{1}{2}, \frac{5}{2})$

{where $[.]$ denotes GIF}

Important Questions on Continuity and Differentiability

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 36

Test the continuity & differentiability of the function defined as under at $x = 1 & x = 2 .$

$f (x) = \{\begin{matrix} x & ; & x < 1 \\ 2 - x & ; & 1 \leq x \leq 2 \\ - 2 + 3 x - x^{2} & ; & x > 2 \end{matrix}\}$

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 38

Prove that

f (x) = |x| \cos (x)

is not differentiable at

x = 0 .

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 39

Show that the function $f (x) = \{\begin{cases} x^{m} \sin (\frac{1}{x}) \\ 0 \end{cases} \begin{matrix} ; & x > 0 \\ ; & x = 0 \end{matrix}$ is,

$(i)$ Differentiable at $x = 0,$ if $m > 1$
$(ii)$ Continuous but not differentiable at $x = 0,$ if $0 < m \leq 1$
$(iii)$ Neither continuous nor differentiable, if $m \leq 0 .$

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Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 40

Examine the differentiability of

f (x) = \sqrt{1 - e^{- x^{2}}}

at

x = 0 .

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 41

If

f (x) = \{\begin{cases} a x^{2} - b & if |x| < 1 \\ - \frac{1}{|x|} & if |x| \geq 1 \end{cases}

is derivable at

x = 1 .

Find the values of

a and b .

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 42

Draw a graph of the function,

y = [x] + |1 - x|, - 1 \leq x \leq 3

. Determine the points, if any, where this function is not differentiable, where

[\cdot]

denotes the greatest integer function.

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 43

Discuss the continuity & derivability of

f (x) = \{\begin{matrix} |x - \frac{1}{2}|; & 0 \leq x < 1 \\ x [x]; & 1 \leq x \leq 2 \end{matrix}

, where

[x]

indicates the greatest integer

x

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 44

Discuss continuity and differentiability of $y = f (x)$ in $[- 2, 5]$ , where [ .] denotes $GIF$ {.} denotes $FPF$

$f (x) = \{\begin{matrix} [x] & , & x \in [- 2, 0] \\ {x} & , & x \in (0, 2) \\ \frac{x^{2}}{4} & , & x \in [2, 3] \\ \frac{1}{\log_{4} (x - 3)} & , & x \in [3, 5] \end{matrix}$