If f(x) = mx + c, f(0) = f'(0) = 1, then f(2) = ___________________.

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Solution

Given f(x) = mx + c
Since f(0) = 1
⇒ f(0) = m(0) + c
i.e 1 = m(0) + c
⇒ c = 1
i.e f(x) = mx + 1
∴ f'(x) = (mx + 1)'
= (mx)' +(1)'
= m(x)' +(1)'
= m × 1 + 0
i.e f'(x) = m
also, given f'(0) = 1
⇒ f'(0) = m
i.e 1 = m
∴ f(x) = x + 1
i.e f(2) = 2 + 1 = 3

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If f(x) = mx + c, f(0) = f'(0) = 1, then f(2) = ___________________.

Given f(x) = mx + c Since f(0) = 1 ⇒ f(0) = m(0) + c i.e 1 = m(0) + c ⇒ c = 1 i.e f(x) = mx + 1 ∴ f'(x) = (mx + 1)' = (mx)' +(1)' = m(x)' +(1)' = m × 1 + 0 i.e f'(x) = m also, given f'(0) = 1 ⇒ f'(0) = m i.e 1 = m ∴ f(x) = x + 1 i.e f(2) = 2 + 1 = 3

Given f(x) = mx + c
Since f(0) = 1
⇒ f(0) = m(0) + c
i.e 1 = m(0) + c
⇒ c = 1
i.e f(x) = mx + 1
∴ f'(x) = (mx + 1)'
= (mx)' +(1)'
= m(x)' +(1)'
= m × 1 + 0
i.e f'(x) = m
also, given f'(0) = 1
⇒ f'(0) = m
i.e 1 = m
∴ f(x) = x + 1
i.e f(2) = 2 + 1 = 3