If `f(x)={{:(e^(2x^(2)+x),":",xgt0),(ax+b,":",xle0):}` is differentiable at `x=0`, then

Options

A. `a=1, b=-1`
B. `a=-1, b=1`
C. `a=1, b=1`
D. `a=-1, b=-1`

Answer

To determine the values of constants \( a \) and \( b \) such that the function

\[ f(x) = \begin{cases} e^{2x^2 + x} & \text{if } x > 0 \\ ax + b & \text{if } x \leq 0 \end{cases} \]

is differentiable at \( x = 0 \), we need to ensure that both the function is continuous at \( x = 0 \) and that the left-hand and right-hand derivatives at \( x = 0 \) are equal.

### Step 1: Ensure Continuity at \( x = 0 \)

For \( f(x) \) to be continuous at \( x = 0 \), we need:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) \]

Calculating \( f(0) \):

\[ f(0) = a(0) + b = b \]

Calculating the left-hand limit as \( x \to 0^- \):

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (ax + b) = b \]

Calculating the right-hand limit as \( x \to 0^+ \):

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} e^{2x^2 + x} = e^{0} = 1 \]

Setting the left-hand limit equal to the right-hand limit for continuity:

\[ b = 1 \]

### Step 2: Ensure Differentiability at \( x = 0 \)

Next, we need to find the derivatives from both sides and set them equal.

**Left-hand derivative** at \( x = 0 \):

\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{a(h) + 1 - 1}{h} = \lim_{h \to 0^-} \frac{ah}{h} = a \]

**Right-hand derivative** at \( x = 0 \):

\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{e^{2h^2 + h} - 1}{h} \]

To evaluate this limit, we can use L'Hôpital's Rule since it is in the form \( \frac{0}{0} \).

Calculating the derivative of the numerator and denominator:

\[ \text{Numerator: } \frac{d}{dh}(e^{2h^2 + h}) = e^{2h^2 + h} \cdot (4h + 1) \]
\[ \text{Denominator: } \frac{d}{dh}(h) = 1 \]

Thus, we have:

\[ f'(0^+) = \lim_{h \to 0^+} e^{2h^2 + h} (4h + 1) = e^{0}(0 + 1) = 1 \]

### Step 3: Set the Derivatives Equal

Setting the left-hand and right-hand derivatives equal gives:

\[ a = 1 \]

### Conclusion

Thus, the values of \( a \) and \( b \) are:

\[ a = 1, \quad b = 1 \]

### Final Answer

The values of \( a \) and \( b \) are both \( 1 \).

---