Question c.Thanks **Mathematics Problem: Function Continuity and Differentiability**Let \( f \) be the function defined as follows:\[f(x) = \begin{cases} |x-1|+2, & \text{for } x < 1 \\ ax^2 + bx, & \text{for } x \geq 1 \end{cases}\]where \( a \) and \( b \) are constants.**Questions:**a) If \( a = 2 \) and \( b = 3 \), is \( f \) continuous for all \( x \)? Justify your answer.b) Describe all values of \( a \) and \( b \) for which \( f \) is a continuous function.c) For what values of \( a \) and \( b \) is \( f \) both continuous and differentiable?

Name: Question c.Thanks **Mathematics Problem: Function Continuity and Diffe
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Question c.Thanks **Mathematics Problem: Function Continuity and Differentiability**Let \( f \) be the function defined as follows:\[f(x) = \begin{cases} |x-1|+2, & \text{for } x < 1 \\ ax^2 + bx, & \text{for } x \geq 1 \end{cases}\]where \( a \) and

Given function is, fx=x-1+2for x<1ax2+bxfor x≥1 The objective is to find the values of a and b for which the function is both continuous and differentiable. Consider the function as, fx=x-1+2for x<1ax2+bxfor x≥1 If the function is continuous then, x-1+2=ax2+bx At x=1, 1-1+2=a12+b1a+b=2 ...... (1) Now, derivative of ax2+bx is 2ax+b At x=1, the derivative is f'1=2a+b And for x<1, fx=x-1+2=-x-1+2=3-x Then derivative is f'x=ddx3-x=-1 For differentiable function, 2a+b=-1 .....(2) On solving equations (1) and (2), a=-3 and b=5 Therefore, for a=-3 and b=5 the function fx=x-1+2for x<1ax2+bxfor x≥1 is continuous and differentiable.