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Question

Let f(x)=x-1+aifx12x+3ifx>1. If f(x) has local minimum at x=1, then:


A

7

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B

8

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C

9

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D

a5

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Solution

The correct option is D

a5


Explanation for the correct option:

Find the value of 'a':

Given,

If x1,f(x)=x-1+a,

Then

If x=-1, f(x)=-1-1+a=2+a

If x=0, f(x)=0-1+a=1+a

If x=1, f(x)=1-1+a=0+a

So, as the value of x increases, the value of the function will decrease. therefore, f(x) is a decreasing function. at x1

Similarly,

If x>1,f(x)=2x+3,

clearly, f(x)=2x+3 is an increasing function at x>1

f(x) has local minima at x=1[Given]

We know that,

f(1)limx1+f(x)

Therefore,

f(1)limh0f(1+h)alimh02(1+h)+3f(x)=2x+3;f(1)=aalimh05+2ha5

Hence, the correct answer is Option (D).


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